Bunuel DS Questions with Explanations
DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard
deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and
tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!
DS Questions
1. An integer greater than 1 that is not prime is called composite. If the two-digit integer n is
greater than 20, is n composite?
(1) The tens digit of n is a factor of the units digit of n.
(2) The tens digit of n is 2.
Given:
--> two digit integer can be written as follows:
-->
,
.
(1) tells that
,(
) -->
will always be composite and factor of . Sufficient
(2) tell that
, but
can be for instance composite
--> as
or prime
,
. Not sufficient.
Answer: A.
2. Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?
(1) Two of the interior angles of ABCD are right angles.
(2) The degree measure of angle ABC is twice the degree measure of angle BCD.
Sum of inner angels of quadrilateral is 360 degrees. (Sum of inner angles of polygon=180(n-2),
where n is # of sides)
(1) Angles can be 90+90 + any combination of two angels totaling 180. Not sufficient.
(2) <ABC=2<BCD. Not sufficient
(1)+(2) Angles can be 90+90+45+135 Or 90+90+60+120 Not sufficient.
Answer: E.
3. Is x + y < 1 ?
(1) x < 8/9
(2) y < 1/8
(1) Info only about x. Not sufficient
(2) Info only about y. Not sufficient
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Bunuel DS Questions with Explanations
(1)+(2) x+y<73/72 Not sufficient
Answer: E.
4. Is x^4 + y^4 > z^4 ?
(1) x^2 + y^2 > z^2
(2) x+y > z
Plugging numbers from Pythagorean triplets is the best way to get that not sufficient.
Answer E.
5. At a certain theater, the cost of each adult's ticket is $5 and the cost of each child's ticket is
$2. What was the average cost of all the adult's and children's tickets sold at the theater
yesterday?
(1) Yesterday ratio of # of children's ticket sold to the # of adult's ticketr sold was 3 to 2
(2) Yesterday 80 adult's tickets were sold at the theater.
Av. cost=(2*C+5*A)/(C+A)
(1) 3A=2C A=2C/3 --> Av.cost C(2+5*2/3)/C(1+2/3) --> (2+5*2/3)/(1+2/3) Sufficient
(2) A=80 know nothing about C Not sufficient.
Answer: A
6. Are some goats not cows?
(1) All cows are lions
(2) All lions are goats.
This is good one:
Question generally asks is g>c?
(1) c<=l Not sufficient
(2) l<=g Not sufficient
(1)+(2) c<=l<=g --> If all cows are lions and all lions are goats there are no goat, which are not
cows, in other case there are, so Not sufficient
Answer: E.
7. Patrick is cleaning his house in anticipation of the arrival of guests. He needs to vacuum the
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Bunuel DS Questions with Explanations
floors, fold the laundry, and put away the dishes after the dishwasher completes its cycle. If the
dishwasher is currently running and has 55 minutes remaining in its cycle, can Patrick complete
all of the tasks before his guests arrive in exactly 1 hour?
(1) Vacuuming the floors and folding the laundry will take Patrick 36 minutes.
(2) Putting away the dishes will take Patrick 7 minutes.
(1) Don't know how much time is needed to put away dishes. Not sufficient
(2) If dishwasher will stop after 55 min and 7 min is needed to put away dishes 55+7=62>60, so
Patrick won't complete all of the tasks before his guests arrive in exactly 1 hour. Sufficient
Answer: B
8. Are all of the numbers in a certain list of 15 numbers equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.
(1) S=60 list can contain numerous combination of 15 numbers totaling 60. Not sufficient.
(2) If the sum of ANY 3 numbers=12 all numbers=12/3=4. Sufficient.
Answer: B.
http://gmatclub.com/forum/collection-of-8-ds-questions-85290.html
DS Questions New:
1. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is even
(2) n < 9
(1) n=2 --> 22+23=45, n=4 --> n=6 x1+(x1+1)+(x1+2)+(x1+3)+(x1+4)+(x1+5)=45 x1=5. At least
two options for n. Not sufficient.
(2) n<9 same thing not sufficient.
(1)+(2) No new info. Not sufficient.
Answer: E.
2. Is a product of three integers XYZ a prime?
(1) X=-Y
(2) Z=1
(1) x=-y --> for xyz to be a prime z must be -p AND x=-y shouldn't be zero. Not sufficient.
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Bunuel DS Questions with Explanations
(2) z=1 --> Not sufficient.
(1)+(2) x=-y and z=1 --> x and y can be zero, xyz=0 not prime OR xyz is negative, so not
prime. In either case we know xyz not prime.
Answer: C
3. Multiplication of the two digit numbers wx and cx, where w,x and c are unique non-zero
digits, the product is a three digit number. What is w+c-x?
(1) The three digits of the product are all the same and different from w c and x.
(2) x and w+c are odd numbers.
(1) wx+cx=aaa (111, 222, ... 999=37*k) --> As x is the units digit in both numbers, a can be
1,4,6 or 9 (2,3,7 out because x^2 can not end with 2,3, or 7. 5 is out because in that case x
also should be 5 and we know that x and a are distinct numbers).
1 is also out because 111=37*3 and we need 2 two digit numbers.
444=37*12 no good we need units digit to be the same.
666=37*18 no good we need units digit to be the same.
999=37*27 is the only possibility all digits are distinct except the unit digits of multiples.
Sufficient
(2) x and w+c are odd numbers.
Number of choices: 13 and 23 or 19 and 29 and w+c-x is the different even number.
Answer: A.
4. Is y – x positive?
(1) y > 0
(2) x = 1 – y
Easy one even if y>0 and x+y=1, we can find the x,y when y-x>0 and y-x<0
Answer: E.
5. If a and b are integers, and a not= b, is |a|b > 0?
(1) |a^b| > 0
(2) |a|^b is a non-zero integer
This is tricky |a|b > 0 to hold true: a#0 and b>0.
(1) |a^b|>0 only says that a#0, because only way |a^b| not to be positive is when a=0. Not
sufficient. NOTE having absolute value of variable |a|, doesn't mean it's positive. It's not
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Bunuel DS Questions with Explanations
negative --> |a|>=0
(2) |a|^b is a non-zero integer. What is the difference between (1) and (2)? Well this is the
tricky part: (2) says that a#0 and plus to this gives us two possibilities as it states that it's
integer:
A. -1>a>1 (|a|>1), on this case b can be any positive integer: because if b is negative |a|^b
can not be integer.
OR
B. |a|=1 (a=-1 or 1) and b can be any integer, positive or negative.
So (2) also gives us two options for b. Not sufficient.
(1)+(2) nothing new: a#0 and two options for b depending on a. Not sufficient.
Answer: E.
6. If M and N are integers, is (10^M + N)/3 an integer?
(1) N = 5
(2) MN is even
Note: it's not given that M and N are positive.
(1) N=5 --> if M>0 (10^M + N)/3 is an integer ((1+5)/3), if M<0 (10^M + N)/3 is a fraction
((1/10^|M|+5)/3). Not sufficient.
(2) MN is even --> one of them or both positive/negative AND one of them or both even.
Not sufficient
(1)+(2) N=5 MN even --> still M can be negative or positive. Not sufficient.
Answer: E.
7. If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value
of c?
(1) d = 3
(2) b = 6
Note this part: "for all values of x"
So, it must be true for x=0 --> c=d^2 --> b=2d
(1) d = 3 --> c=9 Sufficient
(2) b = 6 --> b=2d, d=3 --> c=9 Sufficient
Answer: D.
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Bunuel DS Questions with Explanations
8. If x and y are non-zero integers and |x| + |y| = 32, what is xy?
(1) -4x - 12y = 0
(2) |x| - |y| = 16
(1) x+3y=0 --> x and y have opposite signs --> either 4y=32 y=8 x=-3, xy=-24 OR -4y=32 y=-8
x=3 xy=24. The same answer. Sufficient.
(2) Multiple choices. Not sufficient.
Answer: A.
9. Is the integer n odd
(1) n is divisible by 3
(2) 2n is divisible by twice as many positive integers as n
(1) 3 or 6. Clearly not sufficient.
(2) TIP:
When odd number n is doubled, 2n has twice as many factors as n.
Thats because odd number has only odd factors and when we multiply n by two we remain
all these odd factors as divisors and adding exactly the same number of even divisors, which
are odd*2.
Sufficient.
Answer: B.
10. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is odd
(2) n >= 9
Look at the Q 1 we changed even to odd and n<9 to n>=9
(1) not sufficient see Q1.
(2) As we have consecutive positive integers max for n is 9: 1+2+3+...+9=45. (If n>9=10 first
term must be zero. and we are given that all terms are positive) So only case n=9. Sufficient.
Answer: B.
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Bunuel DS Questions with Explanations
1. When the positive integer x is divided by 4, is the remainder equal to 3?
(1) When x/3 is divided by 2, the remainder is 1.
(2) x is divisible by 5.
Answer: E.
2. In 2003 Acme Computer priced its computers five times higher than its printers. What is the
ratio of its gross revenue for computers and printers respectively in the year 2003?
(1) In the first half of 2003 it sold computers and printers in the ratio of 3:2, respectively, and in
the second half in the ratio of 2:1.
(2) It sold each computer for $1000.
Answer: E
3. Last Tuesday a trucker paid $155.76, including 10 percent state and federal taxes, for diesel
fuel. What was the price per gallon for the fuel if the taxes are excluded?
(1) The trucker paid $0.118 per gallon in state and federal taxes on the fuel last Tuesday.
(2) The trucker purchased 120 gallons of the fuel last Tuesday.
Answer: D.
4. What is the remainder when the positive integer x is divided by 8?
(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.
Answer: E.
5. Al, Pablo, and Marsha shared the driving on a 1500 mile trip, which of the three drove the
greatest distance on the trip?
(1) Al drove 1 hour longer than Pablo but at an average of 5 miles per hour slower than Pablo.
(2) Marsha drove 9 hours and averaged 50 miles per hour
Answer: E.
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Bunuel DS Questions with Explanations
6. How many perfect squares are less than the integer d?
(1) 23 < d < 33
(2) 27 < d < 37
Answer: B.
7. The integers m and p are such that 2 is less than m and m is less than p. Also, m is not a
factor of p. If r is the remainder when p is divided by m, is r > 1.
(1) The greatest common factor of m and p is 2.
(2) The least common multiple of m and p is 30.
Answer: A.
8. A scientist is studying bacteria whose cell population doubles at constant intervals, at which
times each cell in the population divides simultaneously. Four hours from now, immediately
after the population doubles, the scientist will destroy the entire sample. How many cells will
the population contain when the bacteria is destroyed?
(1) Since the population divided two hours ago, the population has quadrupled, increasing by
3,750 cells.
(2) The population will double to 40,000 cells with one hour remaining until the scientist
destroys the sample.
Answer: C.
9. Is x^2 equal to xy?
(1) x^2 - y^2 = (x+5)(y-5)
(2) x=y
Answer: B.
10. A number of oranges are to be distributed evenly among a number of baskets. Each basket
will contain at least one orange. If there are 20 oranges to be distributed, what is the number
of oranges per basket?
(1) If the number of baskets were halved and all other conditions remained the same, there
would be twice as many oranges in every remaining basket.
(2) If the number of baskets were doubled, it would no longer be possible to place at least one
orange in every basket.
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Bunuel DS Questions with Explanations
Answer: B.
11. If p is a prime number greater than 2, what is the value of p?
(1) There are a total of 100 prime numbers between 1 and p+1
(2) There are a total of p prime numbers between 1 and 3912.
Answer: D.
12. If x is a positive integer, what is the least common multiple of x, 6, and 9?
(1) The LCM of x and 6 is 30.
(2) The LCM of x and 9 is 45.
Answer: D.
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Standard Deviation:
Please note the following:
A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD,
but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding
the concept.
B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on
combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's
better to concentrate on issues you'll definitely face on G-day.
Many questions below are easy, some are tough, but anyway they are good to master in solving SD
problems. I'll post OA after some discussions. Please provide your way of thinking along with the
answer. Thanks.
1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year
was $2.3 million.
(2) The standard deviation of Company R’s earnings per month in the second half of this
year was $3.9 million.
Answer: E.
2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.
Answer: A.
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Bunuel DS Questions with Explanations
3. Lifetime of all the batteries produced by certain companies have a distribution which is
symmetric about mean m. If the distribution has a standard deviation of d , what
percentage of distribution is greater than m+d?
(1) 68 % of the distribution in the interval from m-d to m+d, inclusive
(2) 16% of the distribution is less than m-d
Answer: D.
4. Question deleted
5. List S and list T each contain 5 positive integers, and for each list the average of the
integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard
deviation of the integers in list S greater than the standard deviation of the integers in list
T?
(1)The integer 25 is in list S
(2)The integer 45 is in list T
Answer: C.
6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?
(1) All members of T are positive
(2) T consists of only one member
Answer: B.
7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero?
(1) The range of set X is equal to 3
(2) The mean of set X is equal to 5
Answer: A.
8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.
Answer: D.
9. Question deleted
10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the
standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.
Answer: B.
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Bunuel DS Questions with Explanations
11. During an experiment, some water was removed from each of the 6 water tanks. If the
standard deviation of the volumes of water in the tanks at the beginning of the
experiment was 10 gallons, what was the standard deviation of the volumes of water in
the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the
experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the
experiment was 63 gallons.
Answer: A.
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Inequality and absolute value.
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0
First let's simplify given expression
:
-->
. Note here that we CAN NOT reduce this
expression by , as some of you did. Remember we are asked to determine the value of
, and when reducing by you are assuming that doesn't equal to . We don't know
that.
Next: we can conclude that either
or/and
when y=0 and x any value (including 3), OR
(1)
. Which means that
equals to 0,
when y is not equal to zero, and x=3.
. If y is not 0, x must be 3 and y-x to be 3, y must be 6. In this case
But if y=0 then x=-3 and
.
. Two possible scenarios. Not sufficient.
OR:
-->
or
-->
--> if
--> either
, then
and
if
, then
and
. Two different answers. Not sufficient.
(2)
. x is negative, hence x is not equals to 3, hence y must be 0. So, xy=0. Sufficient.
Answer: B.
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Bunuel DS Questions with Explanations
2. 2. If y is an integer and y = |x| + x, is y = 0?
(1) x < 0
(2) y < 1
Note: as
then
is never negative. For
then (when x is negative or zero) then
then
and for
.
(1)
-->
. Sufficient.
(2)
, as we concluded y is never negative, and we are given that
is an integer, hence
. Sufficient.
Answer: D.
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a
(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.
(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.
(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be
0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not
sufficient.
Answer: E.
4. 4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1
(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR
consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this
line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.
(2) x/y>1 --> x and y have the same sign. But we don't know whether they are both positive
or both negative. Not sufficient.
(1)+(2) Again it can be done with different approaches. You should just find the one which is
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Bunuel DS Questions with Explanations
the less time-consuming and comfortable for you personally.
One of the approaches:
-->
-->
too. Sufficient.
--> substitute x -->
-->
is positive, and as
,
is positive
Answer: C.
5. 5. What is the value of y?
(1) 3|x^2 -4| = y - 2
(2) |3 - y| = 11
(1) As we are asked to find the value of y, from this statement we can conclude only that
y>=2, as LHS is absolute value which is never negative, hence RHS als can not be negative.
Not sufficient.
(2) |3 - y| = 11:
y<3 --> 3-y=11 --> y=-8
y>=3 --> -3+y=11 --> y=14
Two values for y. Not sufficient.
(1)+(2) y>=2, hence y=14. Sufficient.
Answer: C.
6. 6. If x and y are integer, is y > 0?
(1) x +1 > 0
(2) xy > 0
(1) x+1>0 --> x>-1. As x is an integer x can take the following values 0,1,2,... But we know
nothing about y. Not sufficient.
(2) xy>0. x and y have the same sign (both positive OR both negative) and neither x nor y is
zero. Not sufficient.
(1)+(2) x is positive, as from (1) it's 0,1,2.. and from (2) x is not zero. Hence xy to be positive
y also must be positive. Sufficient.
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materials. ---ASAX
Bunuel DS Questions with Explanations
Answer: C.
____________
7. 7. |x+2|=|y+2| what is the value of x+y?
(1) xy<0
(2) x>2 y<2
This one is quite interesting.
First note that |x+2|=|y+2| can take only two possible forms:
A. x+2=y+2 --> x=y. This will occur if and only x and y are both >= than -2 OR both <= than -2.
In that case x=y. Which means that their product will always be positive or zero when
x=y=0.
B. x+2=-y-2 --> x+y=-4. This will occur when either x or y is less then -2 and the other is more
than -2.
When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have
scenario B and x+y=-4. Also note that vise-versa is not right. Meaning that we can have
scenario B and xy may be positive as well as negative.
(1) xy<0 --> We have scenario B, hence x+y=-4. Sufficient.
(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B,
hence x+y=-4. Sufficient.
Answer: D.
8. 8. a*b#0. Is |a|/|b|=a/b?
(1) |a*b|=a*b
(2) |a|/|b|=|a/b|
|a|/|b|=a/b is true if and only a and b have the same sign, meaning a/b is positive.
(1) |a*b|=a*b, means a and b are both positive or both negative, as LHS is never negative
(well in this case LHS is positive as neither a nor b equals to zero). Hence a/b is positive in
any case. Hence |a|/|b|=a/b. Sufficient.
(2) |a|/|b|=|a/b|, from this we can not conclude whether they have the same sign or not.
Not sufficient.
Answer: A.
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materials. ---ASAX
Bunuel DS Questions with Explanations
9. 9. Is n<0?
(1) -n=|-n|
(2) n^2=16
(1) -n=|-n|, means that either n is negative OR n equals to zero. We are asked whether n is
negative so we can not be sure. Not sufficient.
(2) n^2=16 --> n=4 or n=-4. Not sufficient.
(1)+(2) n is negative OR n equals to zero from (1), n is 4 or -4 from (2). --> n=-4, hence it's
negative, sufficient.
Answer: C.
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n
Question basically asks is -4<n<4 true.
(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.
(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question.
Not sufficient.
Answer: A.
11. Is |x+y|>|x-y|?
(1) |x| > |y|
(2) |x-y| < |x|
To answer this question you should visualize it. We have comparison of two absolute values.
Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same
sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they
have the same sign will contribute to each other and x-y will not.
5+3=8 and 5-3=2
OR -5-3=-8 and -5-(-3)=-2.
So if we could somehow conclude that x and y have the same sign or not we would be able
to answer the question.
(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.
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materials. ---ASAX
Bunuel DS Questions with Explanations
(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and
origin. This can only happen when x and y have the same sign, when they are both positive
or both negative, when they are at the same side from the origin. Sufficient. (Note that viseversa is not right, meaning that x and y can have the same sign but |x| can be less than |xy|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)
Answer: B.
11. 12. Is r=s?
(1) -s<=r<=s
(2) |r|>=s
This one is tough.
(1) -s<=r<=s, we can conclude two things from this statement:
A. s is either positive or zero, as -s<=s;
B. r is in the range (-s,s) inclusive, meaning that r can be -s as well as s.
But we don't know whether r=s or not. Not sufficient.
(2) |r|>=s, clearly insufficient.
(1)+(2) -s<=r<=s, s is not negative, |r|>=s --> r>=s or r<=-s. This doesn't imply that r=s, from
this r can be -s as well.
Consider: s=5, r=5 --> -5<=5<=5 |5|>=5
s=5, r=-5 --> -5<=-5<=5 |-5|>=5
Both statements are true with these values. Hence insufficient.
Answer: E.
_________________
12. 13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0
Last one.
Is |x-1| < 1? Basically the question asks is 0<x<2 true?
(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This
is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.
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materials. ---ASAX
Bunuel DS Questions with Explanations
(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.
(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range,
thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.
Answer: E.
http://gmatclub.com/forum/inequality-and-absolute-value-questions-from-my-collection86939-40.html
Exponents and Roots:
1. If
?
, where
and
are positive integers, what is the units digit of
(1)
(2)
(1)
--> since both and are positive integers then
and
are perfect squares --> there are only two perfect squares in the given range 121=11^2
and 144=12^2 -->
and
. Sufficient.(As cyclicity of units digit of in integer
power is , therefore the units digit of
(2)
is the same as the units digit of
-->
positive integers then:
--> since both
and
-->
and
, so 3).
and
are
. Sufficient.
Answer: D.
2. 2. If x, y, and z are positive integers and
. Is
(1) is an even perfect square and is an odd perfect cube.
(2) is not an integer.
and integer?
Note: a perfect square, is an integer that can be written as the square of some other
integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer
that can be written as the cube of some other integer. For example 27=3^3, is a
perfect cube.
Make prime factorization of 2,700 -->
(1)
.
is an even perfect square and is an odd perfect cube --> if
and
then
is either
or
must be a perfect square
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materials. ---ASAX
Bunuel DS Questions with Explanations
which makes
then
(2)
an integer:
could be
or
which makes
. But if
not an integer. Not sufficient.
is not an integer. Clearly insufficient.
(1)+(2) As from (1)
then
perfect square which makes
, therefore it must be
an integer:
or
(from 1) -->
is a
. Sufficient.
Answer: C.
3. 3. If
then what is the value of
?
(1)
(2)
--> factor out
from the nominator and apply
the expression in the denominator:
the value of
.
(1)
. So we should find
-->
(note that since
Sufficient.
(2)
to
-->
then the second solution
-->
is not valid).
. Not sufficient.
Answer: A.
4. 4. If
is
?
(1)
(2)
means that neither of unknown is equal to zero. Next,
, so the question becomes: is
and
are positive numbers then the question boils down whether
the same as whether
? Since
, which is
(recall that odd roots have the same sign as the base of the
All contents have been taken form www.gmatclub.com . I do not own the copyright of the any of the
materials. ---ASAX
Bunuel DS Questions with Explanations
root, for example:
(1)
and
).
--> as even root from positive number (
, (or which is the same
is NO. Sufficient.
in our case) is positive then
). Therefore answer to the original question
(2)
--> the same here as
then
, (or which is the same
). Therefore answer to the original question is NO. Sufficient.
Answer: D.
5. 5. If
and
are negative integers, then what is the value of
?
(1)
(2)
(1)
--> as both
and
are negative integers then
-->
or
. Note that as negative
integer (x) in negative integer power (y) gives positive number (1/81) then the power
must be negative even number. Not sufficient.
(2)
--> as the result is negative then
-->
(1)+(2) Only one pair of negative integers
and
-->
must be negative odd number -->
or
and
. Not sufficient.
satisfies both statements
. Sufficient.
Answer: C.
6. 6. If
then what is the value of
?
(1)
(2)
(1)
and
-->
--> applying
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materials. ---ASAX
Bunuel DS Questions with Explanations
we'll get:
-->
-->
, since given that
-->
then
hence
.
Sufficient.
(2)
and
when
--> since
-->
and
then the only case
to hold true is
. Sufficient.
Answer: D.
7. 7. If
is a positive integer is
(1)
is an integer
(2)
is not an integer
an integer?
Must know for the GMAT: if is a positive integer then
is either a positive integer
itself or an irrational number. (It can not be some reduced fraction eg 7/3 or 1/2)
Also note that the question basically asks whether
(1)
is an integer -->
is a perfect square.
can not be a perfect square because if it is, for example if
for some positive integer
Sufficient.
then
(2)
is not an integer -->
Sufficient.
.
-->
.
Answer: D.
8. 8. What is the value of
?
(1)
(2)
(1)
(2)
-->
-->
--->
-->
-->
. Sufficient.
-->
-->
(the power of 3 must be zero in order this equation to hold true) -->
the sum of two non-negative values is zero --> both
and
must be
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materials. ---ASAX
Bunuel DS Questions with Explanations
zero -->
-->
. Sufficient.
Answer: D.
9. 9. If
,
and are non-zero numbers, what is the value of
?
(1)
(2)
(1)
--> infinitely many combinations of , and are possible which will
give different values of the expression in the stem: try x=y=1 and y=-6 or x=1, y=2, z=-3.
Not sufficient.
(2)
-->
--> substitute this value of x into the expression in
the stem --> , as
Sufficient.
then:
.
Must know for the GMAT:
and
.
Answer: B.
10. 10. If and
integer?
are non-negative integers and
is
an even
(1)
(2)
(1)
-->
-->
Since both
and
--> equate the powers:
.
and
are integers (and
-->
) then
and
-->
, so the answer to the question is
No. Sufficient. (Note that
and
-->
valid scenario (solution) as both unknowns must be integers)
and
is not a
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materials. ---ASAX
Bunuel DS Questions with Explanations
(2)
--> obviously
must be an integer (since
) and as
then the only solution is
-->
A.
and
. So,
(notice that
holds true) -->
in this case:
B.
--> two scenarios are possible:
, and
;
and
(notice that
holds true) -->
in this case:
, and
.
Two different answers. Not sufficient.
Answer: A.
11. 11. What is the value of
?
(1)
(2)
Notice that we are not told that the
(1)
--> if
and
and
are integers.
are integers then as
BUT if they are not, then for any value of
then
there will exist some non-integer
to satisfy given expression and vise-versa (for example if
-->
and
then
-->
). Not
sufficient.
(2)
(including 2) or
(1)+(2) If from (2)
then from (1)
Sufficient.
-->
--> either
and
and
is ANY number (including 1). Not sufficient.
then from (1)
-->
-->
. Thus
is ANY number
and if from (2)
and
.
Answer: C.
http://gmatclub.com/forum/tough-and-tricky-exponents-and-roots-questions-12596720.html
Discreet Chart of DS:
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materials. ---ASAX
Bunuel DS Questions with Explanations
1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y
hours. They start working simultaneously and independently at their respective
constant rates at 9:45am. If both x and y are odd integers, is x=y?
(1) x^2+y^2<12
(2) Bonnie and Clyde complete the painting of the car at 10:30am
Solution: the-discreet-charm-of-the-ds-126962-20.html#p1039633
2. Is xy<=1/2?
(1) x^2+y^2=1
(2) x^2-y^2=0
Solution: the-discreet-charm-of-the-ds-126962-20.html#p1039634
3. If a, b and c are integers, is abc an even integer?
(1) b is halfway between a and c
(2) a = b - c
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039637
4. How many numbers of 5 consecutive positive integers is divisible by 4?
(1) The median of these numbers is odd
(2) The average (arithmetic mean) of these numbers is a prime number
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039645
5. What is the value of integer x?
(1) 2x^2+9<9x
(2) |x+10|=2x+8
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039650
6. If a and b are integers and ab=2, is a=2?
(1) b+3 is not a prime number
(2) a>b
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039651
7. A certain fruit stand sold total of 76 oranges to 19 customers. How many of them
bought only one orange?
(1) None of the customers bought more than 4 oranges
(2) The difference between the number of oranges bought by any two customers is even
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materials. ---ASAX
Bunuel DS Questions with Explanations
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039655
8. If x=0.abcd, where a, b, c and d are digits from 0 to 9, inclusive, is x>7/9?
(1) a+b>14
(2) a-c>6
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039662
9. If x and y are negative numbers, is x<y?
(1) 3x + 4 < 2y + 3
(2) 2x - 3 < 3y - 4
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039665
10. The function f is defined for all positive integers a and b by the following rule:
f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If
f(10,x)=11, what is the value of x?
(1) x is a square of an integer
(2) The sum of the distinct prime factors of x is a prime number.
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039671
11. If x and y are integers, is x a positive integer?
(1) x*|y| is a prime number.
(2) x*|y| is non-negative integer.
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039678
12. If 6a=3b=7c, what is the value of a+b+c?
(1) ac=6b
(2) 5b=8a+4c
Solution: the-discreet-charm-of-the-ds-126962-40.html#p1039680
Discreet Charm of DS Solutions by Bunuel:
1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y
hours. They start working simultaneously and independently at their respective
constant rates at 9:45am. If both x and y are odd integers, is x=y?
All contents have been taken form www.gmatclub.com . I do not own the copyright of the any of the
materials. ---ASAX
Bunuel DS Questions with Explanations
Bonnie and Clyde when working together complete the painting of the car ins
hours (sum of the rates equal to the combined rate or reciprocal of total time:
since
-->
). Now, if
then the total time would be:
is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....
(1) x^2+y^2<12 --> it's possible
and
but it's also possible that
,
to be odd and equal to each other if
and
(or vise-versa). Not
sufficient.
(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete
the job in 3/4 of an hour (45 minutes), since it's not odd/2 then and are not
equal. Sufficient.
Answer: B.
2. 2. Is xy<=1/2?
(1) x^2+y^2=1. Recall that
(square of any number is more than or
equal to zero) -->
--> since
then:
-->
. Sufficient.
(2) x^2-y^2=0 -->
. Clearly insufficient.
Answer: A.
3. 3. If a, b and c are integers, is abc an even integer?
In order the product of the integers to be even at leas on of them must be even
(1) b is halfway between a and c --> on the GMAT we often see such statement and
it can ALWAYS be expressed algebraically as
leas on of them is be even? Not necessarily:
also possible that for example
, for
. Now, does that mean that at
,
and
, of course it's
and
. Not sufficient.
(2) a = b - c -->
. Since it's not possible that the sum of two odd integers to
be odd then the case of 3 odd numbers is ruled out, hence at least on of them must
be even. Sufficient.
All contents have been taken form www.gmatclub.com . I do not own the copyright of the any of the
materials. ---ASAX
Bunuel DS Questions with Explanations
Answer: B.
4. 4. How many numbers of 5 consecutive positive integers is divisible by 4?
(1) The median of these numbers is odd --> the median of the set with odd number
of terms is just a middle term, thus our set of 5 consecutive numbers is: {Odd, Even,
Odd, Even, Odd}. Out of 2 consecutive even integers only one is a multiple of 4.
Sufficient.
(2) The average (arithmetic mean) of these numbers is a prime number --> in any
evenly spaced set the arithmetic mean (average) is equal to the median -->
mean=median=prime. Since it's not possible that median=2=even, (in this case not
all 5 numbers will be positive), then median=odd prime, and we have the same case
as above. Sufficient.
Answer: D.
5. 5. What is the value of integer x?
(1) 2x^2+9<9x --> factor qudratics:
--> roots are and 3 --> "<"
sign indicates that the solution lies between the roots:
--> since there
only integer in this range is 2 then
. Sufficient.
(2) |x+10|=2x+8 --> LHS is an absolute value, which is always non negative, hence
RHS must also be non-negative:
positive hence
-->
-->
, for this range
-->
is
. Sufficient.
Answer: D.
Check this for more on solving inequalities like the one in the first statement:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863
xy-plane-71492.html?hilit=solving%20quadratic#p841486
6. 6. If a and b are integers and ab=2, is a=2?
Notice that we are not told that a and b are positive.
There are following integer pairs of (a, b) possible: (1, 2), (-1, -2), (2, 1) and (-2, -1).
Basically we are asked whether we have the third case.
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materials. ---ASAX
Bunuel DS Questions with Explanations
(1) b+3 is not a prime number --> rules out 1st and 4th options. Not sufficient.
(2) a>b --> again rules out 1st and 4th options. Not sufficient.
(1)+(2) Still two options are left: (-1, -2) and (2, 1). Not sufficient.
Answer: E.
7. 7. A certain fruit stand sold total of 76 oranges to 19 customers. How many of
them bought only one orange?
(1) None of the customers bought more than 4 oranges --> this basically means that
all customers bought exactly 4 oranges (76/19=4), because if even one customer
bought less than 4, the sum will be less than 76. Hence, no one bought only one
orange. Sufficient.
(2) The difference between the number of oranges bought by any two customers is
even --> in order the difference between ANY number of oranges bought to be
even, either all customers must have bought odd number of oranges or all
customers must have bough even number of oranges. But the first case is not
possible: the sum of 19 odd numbers is odd and not even like 76. Hence, again no
one bought only one=odd orange. Sufficient.
Answer: D.
8. 8. If x=0.abcd, where a, b, c and d are digits from 0 to 9, inclusive, is x>7/9?
First of all 7/9 is a recurring decimal =0.77(7). For more on converting Converting
Decimals to Fractions see: math-number-theory-88376.html
(1) a+b>14 --> the least value of a is 6 (6+9=15>14), so in this case x=0.69d<0.77(7)
but a=7 and b=9 is also possible, and in this case x=0.79d>0.77(7). Not sufficient.
(2) a-c>6 --> the least value of a is 7 (7-0=7>6), but we don't know the value of b.
Not sufficient.
(1)+(2) The least value of a is 7 and in this case from (1) least value of b is 8
(7+8=15>14), hence the least value of x=0.78d>0.77(7). Sufficient.
Answer: C.
___________
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materials. ---ASAX
Bunuel DS Questions with Explanations
9. 9. If x and y are negative numbers, is x<y?
(1) 3x + 4 < 2y + 3 -->
. can be some very small number for instance 100 and some large enough number for instance -3 and the answer would be YES,
BUT if
sufficient.
and
then the answer would be NO,
(2) 2x - 3 < 3y - 4 -->
-->
(as y+negative is "more negative" than y). Sufficient.
. Not
-->
Answer: B.
10. 10. The function f is defined for all positive integers a and b by the following rule:
f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If
f(10,x)=11, what is the value of x?
Notice that the greatest common factor of 10 and x, GCF(10,x), naturally must be a
factor of 10: 1, 2, 5, and 10. Thus from f(10,x)=11 we can get four different values of
x:
GCF(10,x)=1 -->
-->
GCF(10,x)=2 -->
-->
GCF(10,x)=5 -->
-->
GCF(10,x)=10 -->
(1) x is a square of an integer -->
-->
;
;
;
.
can be 1 or 100. Not sufficient.
(2) The sum of the distinct prime factors of x is a prime number ---> distinct primes
of 12 are 2 and 3:
, distinct primes of 45 are 3 and 5:
and distinct primes of 100 are 2 and 5:
can be 12 or 100. Not sufficient.
(1)+(2)
can only be 100. Sufficient.
Answer: C.
_________________
All contents have been taken form www.gmatclub.com . I do not own the copyright of the any of the
materials. ---ASAX
.
Bunuel DS Questions with Explanations
11. 11. If x and y are integers, is x a positive integer?
(1) x*|y| is a prime number --> since only positive numbers can be primes, then:
x*|y|=positive --> x=positive. Sufficient
(2) x*|y| is non-negative integer. Notice that we are told that x*|y| is non-negative,
not that it's positive, so x can be positive as well as zero. Not sufficient.
Answer: A.
12. 12. If 6a=3b=7c, what is the value of a+b+c?
Given:
write:
--> least common multiple of 6, 3, and 7 is 42 hence we ca
, for some number -->
,
and
.
(1) ac=6b -->
-->
(2) 5b=8a+4c -->
-->
-->
-->
-->
or
. Not sufficient.
-->
-->
. Sufficient.
Answer: B.
http://gmatclub.com/forum/the-discreet-charm-of-the-ds-126962-40.html
Devil’s Dozen Questions
1. Jules and Jim both invested certain amount of money in bond M for one year,
which pays for 12% simple interest annually. If no other investment were made,
then Jules initial investment in bond M was how many dollars more than Jim's
investment in bond M.
(1) In one year Jules earned $24 more than Jim from bond M.
(2) If the interest were 20% then in one year Jules would have earned $40 more
than Jim from bond M.
Solution: devil-s-dozen-129312.html#p1063846
2. If n is a positive integer and p is a prime number, is p a factor of n!?
(1) p is a factor of (n+2)!-n!
(2) p is a factor of (n+2)!/n!
Solution: devil-s-dozen-129312.html#p1063847
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materials. ---ASAX
Bunuel DS Questions with Explanations
3. If x and y are integers, is y an even integer?
(1) 4y^2+3x^2=x^4+y^4
(2) y=4-x^2
Solution: devil-s-dozen-129312.html#p1063848
4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the
patients have acrophobia?
(1) The number of patients of Vertigo Hospital who have both arachnophobia and
acrophobia is the same as the number of patients who have neither arachnophobia
nor acrophobia.
(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia.
Solution: devil-s-dozen-129312.html#p1063863
5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how
many of 35 astronauts at Solaris space station listen to Bach?
(1) Of the astronauts who do NOT listen to Bach 56% are male.
(2) Of the astronauts who listen to Bach 70% are female.
Solution: devil-s-dozen-129312.html#p1063867
6. Is the perimeter of triangle with the sides a, b and c greater than 30?
(1) a-b=15.
(2) The area of the triangle is 50.
Solution: devil-s-dozen-129312.html#p1063871
7. Set A consists of k distinct numbers. If n numbers are selected from the set oneby-one, where n<=k, what is the probability that numbers will be selected in
ascending order?
(1) Set A consists of 12 even consecutive integers.
(2) n=5.
Solution: devil-s-dozen-129312-20.html#p1063874
8. If p is a positive integer, what is the remainder when p^2 is divided by 12?
(1) p is greater than 3.
(2) p is a prime.
Solution: devil-s-dozen-129312-20.html#p1063884
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materials. ---ASAX
Bunuel DS Questions with Explanations
9. The product of three distinct positive integers is equal to the square of the
largest of the three numbers, what is the product of the two smaller numbers?
(1) The average (arithmetic mean) of the three numbers is 34/3.
(2) The largest number of the three distinct numbers is 24.
Solution: devil-s-dozen-129312-20.html#p1063886
10. There is at least one viper and at least one cobra in Pandora's box. How many
cobras are there?
(1) There are total 99 snakes in Pandora's box.
(2) From any two snakes from Pandora's box at least one is a viper.
Solution: devil-s-dozen-129312-20.html#p1063888
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45
enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies.
(2) $15 is enough to buy 10 muffins and 8 brownies.
Solution: devil-s-dozen-129312-20.html#p1063892
12. If x>0 and xy=z, what is the value of yz?
(1)
(2)
.
.
Solution: devil-s-dozen-129312-20.html#p1063894
13. Buster leaves the trailer at noon and walks towards the studio at a constant
rate of B miles per hour. 20 minutes later, Charlie leaves the same studio and
walks towards the same trailer at a constant rate of C miles per hour along the
same route as Buster. Will Buster be closer to the trailer than to the studio when
he passes Charlie?
(1) Charlie gets to the trailer in 55 minutes.
(2) Buster gets to the studio at the same time as Charlie gets to the trailer.
Solution: devil-s-dozen-129312-20.html#p1063897
Devil’s Dozen Solutions by Bunuel:
All contents have been taken form www.gmatclub.com . I do not own the copyright of the any of the
materials. ---ASAX
Bunuel DS Questions with Explanations
1. Jules and Jim both invested certain amount of money in bond M for one year,
which pays for 12% simple interest annually. If no other investment were
made, then Jules initial investment in bond M was how many dollars more
than Jim's investment in bond M.
Question:
(1) In one year Jules earned $24 more than Jim from bond M.
-->
-->
. Sufficient.
(2) If the interest were 20% then in one year Jules would have earned $40 more
than Jim from bond M. Basically the same type of information as above:
-->
-->
. Sufficient.
Answer: D.
2. 2. If n is a positive integer and p is a prime number, is p a factor of n!?
(1) p is a factor of (n+2)!-n! --> if
then
then answer will be YES but for
the answer will be NO. Not sufficient.
(2) p is a factor of (n+2)!/n! -->
--> if
and for
answer will be NO. Not sufficient.
(1)+(2)
and for
then
then answer will be YES but for
. Now,
the
and
are consecutive integers. Two consecutive integers are co-prime,
which means that they don't share ANY common factor but 1. For example 20
and 21 are consecutive integers, thus only common factor they share is 1. So, as
from (2)
is a factor of
then it can not be a factor of
, thus in order to be a factor of
from (1), then it should be a factor of the first multiple of this expression:
Sufficient.
,
.
Answer: C.
3. 3. If x and y are integers, is y an even integer?
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Bunuel DS Questions with Explanations
(1) 4y^2+3x^2=x^4+y^4 --> rearrange:
-->
. Notice that LHS is even for any value of
odd then
naturally even. So,
even, since if is odd then
and if
: if
is
is even then the product is
is also even, but in order it to be even
must be
. Sufficient.
(2) y=4-x^2 --> if
then
then
but if
. Not sufficient.
Answer: A.
4.
5. 5. If at least one astronaut do NOT listen to Bach at Solaris space station, then
how many of 35 astronauts at Solaris space station listen to Bach?
Also very tricky.
(1) Of the astronauts who do NOT listen to Bach 56% are male --> if # of
astronauts who do NOT listen to Bach is then
is # of males who do
NOT listen to Bach. Notice that
must be an integer. Hence x
must be a multiple of 25: 25, 50, 75, ... But (# of astronauts who do NOT listen
to Bach) must also be less than (or equal to) 35. So can only be 25, which
makes # of astronauts who do listen to Bach equal to 35-25=10. Sufficient.
(2) Of the astronauts who listen to Bach 70% are female. Now, if we apply the
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Bunuel DS Questions with Explanations
same logic here we get that, if # of astronauts who listen to Bach is
then
is # of females who listen to Bach:
must be an integer. Hence
it must be a multiple of 10, but in this case it can take more than 1 value: 10, 20,
30. So, this statement is not sufficient.
Answer: A.
6. 6. Is the perimeter of triangle with the sides a, b and c greater than 30?
700+ question.
(1) a-b=15. Must know for the GMAT: the length of any side of a triangle must
be larger than the positive difference of the other two sides, but smaller than
the sum of the other two sides. So, a+b>c>15 --> a+b+c>30. Sufficient.
(2) The area of the triangle is 50. For a given perimeter equilateral triangle has
the largest area. Now, if the perimeter were equal to 30 then it would have the
largest area if it were equilateral. Let's find what this area would be:
. Since even
equilateral triangle with perimeter of 30 can not produce the area of 50, then
the perimeter must be more that 30. Sufficient.
Answer: D.
7. 7. Set A consists of k distinct numbers. If n numbers are selected from the set
one-by-one, where n<=k, what is the probability that numbers will be selected
in ascending order?
(1) Set A consists of 12 even consecutive integers;
(2) n=5.
We should understand following two things:
1. The probability of selecting any n numbers from the set is the same. Why
should any subset of n numbers have higher or lower probability of being
selected than some other subset of n numbers? Probability doesn't favor any
particular subset.
2. Now, consider that the subset selected is
, where
. We can select this subset of numbers in
out of these n! ways only one, namely
# of ways and
will be in ascending
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Bunuel DS Questions with Explanations
order. So 1 out of n!.
.
Hence, according to the above the only thing we need to know to answer the
question is the size of the subset (n) we are selecting from set A.
Answer: B.
8. 8. If p is a positive integer, what is the remainder when p^2 is divided by 12?
(1) p is greater than 3.
(2) p is a prime.
(1) p is greater than 3. Clearly insufficient: different values of
different values of the remainder.
(2) p is a prime. Also insufficient: if
then the remainder is 9.
will give
then the remainder is 4 but if
(1)+(2) You can proceed with number plugging and try several prime numbers
greater than 3 to see that the remainder will always be 1 (for example try
,
,
).
If you want to double-check this with algebra you should apply the following
property of the prime number: any prime number greater than 3 can be
expressed either as
or
.
If
then
divided by 12;
which gives remainder 1 when
If
then
divided by 12.
which also gives remainder 1 when
Answer: C.
9. 9. The product of three distinct positive integers is equal to the square of the
largest of the three numbers, what is the product of the two smaller numbers?
700 question.
Let the three integers be , , and , where
. Question:
. Given:
-->
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materials. ---ASAX
Bunuel DS Questions with Explanations
(1) The average (arithmetic mean) of the three numbers is 34/3 -->
-->
are integers, then
-->
and
. Now, since and
.
and
-->
.
Sufficient. (Notice that
and
is not possible since in this case
and we are told that all integers are positive).
(2) The largest number of the three distinct numbers is 24. Directly give the
value of c. Sufficient.
Answer: D.
10. 10. There is at least one viper and at least one cobra in Pandora's box. How
many cobras are there?
Quite tricky.
(1) There are total 99 snakes in Pandora's box. Clearly insufficient.
(2) From any two snakes from Pandora's box at least one is a viper. Since from
ANY two snakes one is a viper then there can not be 2 (or more) cobras and
since there is at least one cobra then there must be exactly one cobra in the
box. Sufficient.
Answer: B.
11. 11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45
enough to buy 27 muffins and 27 brownies?
700+ question.
Given:
, where
brownie respectively.
and are prices of one muffin and one
Question: is
? -->
. Question basically asks
whether we can substitute 2 muffins with 2 brownies.
Now if
we can easily substitute 2 muffins with 2 brownies (since
be more than ). But if
we won't know this for sure.
will
But consider the case when we are told that we can substitute 3 muffins with 3
brownies. In both cases (
or
) it would mean that we can substitute
2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether
we can substitute 4 (so more than 3) muffins with 4 brownies.
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Bunuel DS Questions with Explanations
(1) $15 is enough to buy 7 muffins and 11 brownies -->
: we can
substitute 4 muffins with 4 brownies, so according to above we can surely
substitute 2 muffins with 2 brownies. Sufficient.
(1) $15 is enough to buy 10 muffins and 8 brownies -->
: we can
substitute 1 muffin with 1 brownie, so according to above this is does not
ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.
Answer: A.
12. 12. If x>0 and xy=z, what is the value of yz?
(1)
. If
and
then
and
but if
then
,
. Not sufficient.
(2)
-->
-->
--> since
then:
. Sufficient.
Answer: B.
13. 13. Buster leaves the trailer at noon and walks towards the studio at a
constant rate of B miles per hour. 20 minutes later, Charlie leaves the same
studio and walks towards the same trailer at a constant rate of C miles per
hour along the same route as Buster. Will Buster be closer to the trailer than
to the studio when he passes Charlie?
(1) Charlie gets to the trailer in 55 minutes. No info about Buster. Not sufficient.
(2) Buster gets to the studio at the same time as Charlie gets to the trailer -->
Charlie needed 20 minutes less than Buster to cover the same distance, which
means that the rate of Charlie is higher than that of Buster. Since after they pass
each other they need the same time to get to their respective destinations (they
get at the same time to their respective destinations) then Buster had less
distance to cover ahead (at lower rate) than he had already covered (which
would be covered by Charlie at higher rate). Sufficient.
Answer: B.
1.If r is the remainder when the positive integer n is divided by 7, what is the value of r
1. when n is divided by 21, the remainder is an odd number
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Bunuel DS Questions with Explanations
2. when n is divided by 28, the remainder is 3
The possible reminders can be 1,2,3,4,5 and 6. We have the pinpoint the exact remainder from this 6
numbers.
St 1: when n is divided by 21 ( 7 and 3) the remainder is an odd number.
But it cannot be 7, 3 or 9 . Hence the possibilities are : 1 and 5.
Hence there can be two remainders ,1 and 5, when divided by 7.
NOT SUFFICIENT
St 2: when n is divided by 28 the remainder is 3.
As 7 is a factor of 28, the remainder when divided by 7 will be 3
SUFFICIENT
2 If n and m are positive integers, what is the remainder when 3^(4n + 2 + m) is divided by 10 ?
(1) n = 2
(2) m = 1
The Concept tested here is cycles of powers of 3.
The cycles of powers of 3 are : 3,9,7,1
St I) n = 2. This makes 3^(4*2 +2 + m) = 3^(10+m). we do not know m and hence cannot figure out the
unit digit.
St II) m=1 . This makes 3^(4*n +2 + 1).
4n can be 4,8,12,16...
3^(4*n +2 + 1) will be 3^7,3^11, 3^15,3^19 ..... in each case the unit digit will be 7. SUFF
Hence B
3.If p is a positive odd integer, what is the remainder when p is divided by 4 ?
(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.
st1. take multiples of 8....divide them by 4...remainder =1 in each case...
st2. p is odd ,since p is square of 2 integers...one will be even and other odd....now when we divide any
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materials. ---ASAX
Bunuel DS Questions with Explanations
even square by 4 v ll gt 0 remainder..and when divide odd square vll get 1 as remainder......so intoatal
remainder=1
Ans : D
4.If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15 ?
(1) The remainder when p + n is divided by 5 is 1.
(2) The remainder when p - n is divided by 3 is 1.
Ans: E
st1) p+n=6,11,16....insuff.
st2) p-n=4,7,10....insuff...
multiply these two to get p^2-n^2.....multiplying any ttwo values from the above results in different
remainder......
also can be done thru equation....p+n=5a+1..and so on
5.What is the remainder when the positive integer x is divided by 3 ?
(1) When x is divided by 6, the remainder is 2.
(2) When x is divided by 15, the remainder is 2.
Easy one , answer D
st 1...multiple of 6 will also be multiple of 3 so remainder wil be same as 2.
st2) multiple of 15 will also be multiple of 3....so the no.that gives remaindr 2 when divided by 15 also
gives 2 as the remainder when divided by 3...
6.What is the remainder when the positive integer n is divided by 6 ?
(1) n is a multiple of 5.
(2) n is a multiple of 12.
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Bunuel DS Questions with Explanations
Easy one. Answer B
st 1) multiples of 5=5,10,15....all gives differnt remainders with 6
st2)n is divided by 12...so it will be divided by 6...remainder=0
7If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?
(1) y = 6
(2) z = 3
We need to know all the variables. We cannot get that from both the statements. Hence the answer is E.
8.If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r ?
(1) n + 1 is divisible by 3.
(2) n > 20
Answer A
st1... n+1 divisible by 3..so n=2,5,8,11......
this gives 4+7n=18,39,60....remainder 0 in each case......
st2) insufficient ....n can have any value
9.If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of
r?
(1) n is not divisible by 2.
(2) n is not divisible by 3.
ST 1- if n is not divisible by 2, then n is odd, so both (n - 1) and (n + 1) are even. moreover, since every
other even number is a multiple of 4, one of those two factors is a multiple of 4. so the product (n - 1)(n
+ 1) contains one multiple of 2 and one multiple of 4, so it contains at least 2 x 2 x 2 = three 2's in its
prime factorization.
But this is not sufficient, because it can be (n-1)*(n+1) can be 2*4 where remainder is 8. it can be 4*6
where the remainder is 0.
ST 2- if n is not divisible by 3, then exactly one of (n - 1) and (n + 1) is divisible by 3, because every third
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Bunuel DS Questions with Explanations
integer is divisible by 3. therefore, the product (n - 1)(n + 1) contains a 3 in its prime factorization.
Just like st 1 this is not sufficient
the overall prime factorization of (n - 1)(n + 1) contains three 2's and a 3.
therefore, it is a multiple of 24.
sufficient
Answer C
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