Number Properties Guide – DS Questions
101.
3.3
Assorted Questions
3.3.1
Problem Solving
93
If a2 + b2 + c 2 = ab + bc + ca and a = 5, what is the value of (a + b + c)?
(A)
5
(B)
8
(C)
12
(D)
15
(E)
20
Solve yourself:
102.
For how many integer values of n, is
(A)
Five
(B)
Six
(C)
Seven
(D)
Eight
(E)
Nine
✓
30 + 2n
n
◆
a positive integer?
Solve yourself:
103.
r⇣
q
⌘
3
0.000375 2
3
Which of the following is the value of (0.0004) +
24
(A)
0.000063
(B)
0.000633
(C)
0.000705
(D)
0.006330
(E)
0.007050
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Number Properties Guide – DS Questions
Solve yourself:
104.
If n leaves a remainder 5 when divided by d, and a remainder 11 when divided by 2d,
what is the value of d?
(A)
6
(B)
7
(C)
8
(D)
9
(E)
12
Solve yourself:
105.
What is the smallest three digit number that when divided by any of the integers from 2
to 6, leaves a remainder 1 in each case?
(A)
111
(B)
121
(C)
122
(D)
241
(E)
721
Solve yourself:
106.
If x 2
2y 2 = 0, what is the value of
(A)
p
2
(B)
x
?
y
0
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Number Properties Guide – DS Questions
(C)
(D)
1
p
(E)
2
95
2
Solve yourself:
107.
If
2  x  8 and 2  y  5, what is the maximum value of
(A)
1
(B)
2
(C)
3
(D)
3.5
(E)
4
x
y
y
!
?
Solve yourself:
108.
If x =
z?
p
p
p
3
4
2, y = 4 and z = 10, which of the following is the correct order of x, y and
(A)
x<y <z
(B)
x<z<y
(C)
z>x>y
(D)
z>y >x
(E)
y >z>x
Solve yourself:
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96
Number Properties Guide – DS Questions
109.
What is the smallest value of (m + n), where m and n are positive integers, such that
2n > 56 > 2m ?
(A)
12
(B)
14
(C)
15
(D)
26
(E)
27
Solve yourself:
110.
n
If n is divided by 18, the non-zero remainder is r . If 18
is a terminating decimal, how
many values of r are possible?
(A)
One
(B)
Two
(C)
Three
(D)
Nine
(E)
Seventeen
Solve yourself:
111.
If n is the product of integers from 21 to 30, what is the highest exponent of 3 that can
divide n perfectly?
(A)
1
(B)
2
(C)
6
(D)
8
(E)
14
Solve yourself:
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Number Properties Guide – DS Questions
112.
97
If a and b are the highest possible exponents of 2 and 3, respectively that are factors of
8! and 11! = 2a ⇥ 3b ⇥ k, what is the value of k?
(A)
(B)
(C)
(D)
(E)
15 ⇥ 77
45 ⇥ 154
45 ⇥ 770
90 ⇥ 1, 540
90 ⇥ 7, 700
Solve yourself:
113.
What is the value of
(A)
(B)
1
p
(C)
p
(D)
2
(E)
3
✓
◆
1
1
1
p +p
p +p
?
3+2
1+ 2
2+ 3
2
3
Solve yourself:
114.
If p, q and r are integers such that |p|
non-negative value of p + r ?
(A)
2
(B)
3
(C)
5
(D)
6
(E)
8
|q| = |r | and q
3, what is the minimum
Solve yourself:
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98
115.
Number Properties Guide – DS Questions
If p and q are positive integers, what is the minimum value of (p + q) given that
p
3
< ?
q
4
(A)
3
(B)
5
(C)
7
(D)
12
(E)
13
7
<
10
Solve yourself:
116.
If k is a positive integer, how many values of k exist so that
(A)
None
(B)
One
(C)
Two
(D)
Three
(E)
Five
✓
45
4k 1
◆
is an integer?
Solve yourself:
117.
!
q2 + 24
If p =
, what is the minimum possible integer value of |p|, given that q is a
q
negative integer?
(A)
6
(B)
8
(C)
10
(D)
11
(E)
25
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Number Properties Guide – DS Questions
99
Solve yourself:
118.
37
What is the value of (a + b + c), if
=
16
integers.
(A)
3
(B)
5
(C)
10
(D)
12
(E)
13
a + ⇣ 11⌘
b+ c
!
, where a, b and c are positive
Solve yourself:
119.
If the Least Common Multiple (LCM) of two integers a and b is 120 and their Greatest
Common Divisor (GCD) is 6, how many values exist for (a + b)?
(A)
One
(B)
Two
(C)
Three
(D)
Four
(E)
Five
Solve yourself:
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100
Number Properties Guide – DS Questions
120.
If [x] denotes the greatest integer
less

 than or equal to x, for how many less than 10
x
x
positive integer values of x, is
=
+1 ?
3
10
(A)
One
(B)
Two
(C)
Three
(D)
Four
(E)
Five
Solve yourself:
121.
What is the minimum value of
(A)
(B)
◆
12
, if
|6 2x|
3  x  8?
1
0
(D)
1
2
1
(E)
2
(C)
✓
Solve yourself:
122.
If x and y are numbers satisfying (x
maximum value of |xy|?
(A)
24
(B)
36
(C)
42
(D)
56
(E)
70
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2)2 = 25 and x
y
3
4
= 256, what is the
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Number Properties Guide – DS Questions
101
Solve yourself:
123.
Which of the following is NOT a factor of 212 + 213 + 214 + 215 + 216 + 217 ?
(A)
12
(B)
56
(C)
72
(D)
132
(E)
144
Solve yourself:
124.
How many three digit numbers aba exist, where a and b are non-zero digits in the
hundreds, tens and units places, respectively, so that the number aba is divisible by 4?
(A)
16
(B)
18
(C)
22
(D)
24
(E)
25
Solve yourself:
125.
There are three distinct numbers such that the Greatest Common Divisor (GCD) of each
pair of numbers is 3. If the product of these three numbers is 540, how many such sets
of three numbers are there?
(A)
One
(B)
Two
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102
Number Properties Guide – DS Questions
(C)
Three
(D)
Six
(E)
Nine
Solve yourself:
126.
The Least Common Multiple (LCM) of the first 25 positive integers is L. What is the LCM
of the first 27 positive integers when expressed in terms of L?
(A)
L
(B)
3L
(C)
6L
(D)
39L
(E)
78L
Solve yourself:
127.
What is the units digit of 111! + 212! + 313! ?
(A)
0
(B)
2
(C)
4
(D)
6
(E)
8
Solve yourself:
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Number Properties Guide – DS Questions
128.
What is the sum of all positive integer values of x such that
(A)
2
(B)
6
(C)
8
(D)
12
(E)
14
103
✓
2x + 28
x+6
◆
is an integer?
Solve yourself:
129.
What is the product of all possible values of x such that (2x
(A)
4
(B)
3
(C)
0
(D)
1
(E)
4
1)(x+4) = 1?
Solve yourself:
130.
The sum of digits of a three digit number is 6. What is the maximum value of the product
of the three digits?
(A)
4
(B)
5
(C)
6
(D)
8
(E)
9
Solve yourself:
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104
131.
Number Properties Guide – DS Questions
If x and y are integers and 2x + 2xy = y, how many values of x are possible?
(A)
None
(B)
One
(C)
Two
(D)
Three
(E)
Four
Solve yourself:
132.
⇣
⌘ p
1
If 1+ a+b = 2, what is the maximum possible integer value of a?, where a is a positive
integer and b is any positive number.
(A)
1
(B)
2
(C)
3
(D)
4
(E)
5
Solve yourself:
133.
If a = bc , b = c a and c = a4 , what is the value of ac?, where each of a, b, and c are
greater than 1.
(C)
1
4
1
3
1
2
(D)
1
(E)
4
(A)
(B)
Solve yourself:
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Number Properties Guide – DS Questions
134.
105
How many values of x satisfy x + [x] = 5?, where [x] denotes the greatest integer less
than or equal to x.
(A)
None
(B)
One
(C)
Two
(D)
Three
(E)
Five
Solve yourself:
135.
From 1 to 20, for how many values of n, does (3n2 + 2) and (9n2 + 4) have no common
factors other than 1?
(A)
1
(B)
4
(C)
10
(D)
14
(E)
20
Solve yourself:
136.
A 3-digit number xyz, which when divided by its sum of digits has the minimum value.
What is the value of x + y + z ?, where x, y and z are the digits in the hundreds, tens
and units positions?
(A)
2
(B)
3
(C)
11
(D)
18
(E)
19
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106
Number Properties Guide – DS Questions
Solve yourself:
137.
The two digit number x9, which when divided by its sum of digits leaves a maximum
possible remainder. What is the value of x?, where x and 9 are the digits in the tens and
units positions, respectively.
(A)
1
(B)
2
(C)
4
(D)
7
(E)
8
Solve yourself:
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Number Properties Guide – DS Questions
3.3.2
138.
107
Data Sufficiency
If 8x + y = 26, where x and y are positive integers, what is the value of y?
(1)
(2)
x + y > 10
x + y < 15
Solve yourself:
139.
If x is a non-negative integer and y is an integers, does 5x + 3y = 36?
(1)
y 2 = 36
(2)
5|x| = 3y
Solve yourself:
140.
Is
(1)
(2)
a
an integer?; b 6= 0.
b
|a| = |b|
a and b both are unequal non-integers numbers of the same magnitude.
Solve yourself:
141.
Is p > q?
(1)
(2)
xp + xq
0
xp + xq  0
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108
Number Properties Guide – DS Questions
Solve yourself:
142.
What is the value of x?
(1)
(2)
36
x
(x
4 = 14
4) (x
1) =
x
Solve yourself:
143.
If p = q4 , what is the value of
(1)
(2)
p
?
q
p = 16
pq = 32
Solve yourself:
144.
Is p even?
(1)
p
(2)
2p is even.
7
p + 2 is even.
Solve yourself:
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Number Properties Guide – DS Questions
145.
If a 6= 0, is a =
(1)
(2)
a3 = a
a2 =
109
1?
a
Solve yourself:
146.
How many prime numbers are there between 82 and p?
(1)
83 < p < 96
(2)
90 < p < 99
Solve yourself:
147.
Is pq > 0?
(1)
p 2 q3 < 0
(2)
pq2 < 0
Solve yourself:
148.
If S is a finite set of positive numbers, does the set contain a number equal to the average
of all the numbers in the set?
(1)
All numbers in the set are distinct multiples of 6.
(2)
The set contains an odd number of terms which are consecutive multiples of 6.
Solve yourself:
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110
149.
Number Properties Guide – DS Questions
Is
p
= 1?
q
(1)
p
q
(2)
!2
+
q
p
!2
= 2.
p
q
+ = 2.
q
p
Solve yourself:
150.
What is the value of k?
1)2 = 4k
(1)
k(k
(2)
k is an integer divisible by 3.
Solve yourself:
151.
If the symbol ‘8’ represents one among addition, subtraction, multiplication and division, and r is any non-zero number, what does ‘8’ represent?
(1)
(2)
r 8 0 = r.
(r 8 0) 8 r = 0.
Solve yourself:
152.
p, q and r denote the integer lengths of three line segments. What should be the minimum value of p so that the line segments form a triangle?
(1)
(2)
q =r +6
q = 8, r = 2
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Number Properties Guide – DS Questions
111
Solve yourself:
153.
What is the two-digit number A?
(1)
The tens digit is twice the units digit of A.
(2)
The sum of the digits of A is 6.
Solve yourself:
154.
If m and n are positive integers, is m + n2 odd?
(1)
(2)
mn + n2 is odd.
m2 + n2 is odd.
Solve yourself:
155.
If r , s and t are integers, is (r s + t) even?
(1)
r is 3 times a prime number, greater than 5 and s is not divisible by 2.
(2)
t leaves a remainder 3 when divided by 4.
Solve yourself:
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112
156.
Number Properties Guide – DS Questions
If m is a multiple of an odd prime number p, is m also a multiple of p 2 ?
(1)
p<8
(2)
m = 60
Solve yourself:
157.
What is the value of the expression
x4y 3 x2y 5
?
x2y 3 y 5
(1)
x=2
(2)
xy = 1; where both x and y are integers.
Solve yourself:
158.
Is
m
= 1?
n
(1)
m
n
=
n
m
(2)
(m
n)2 = m2
n2
Solve yourself:
159.
Is abcd = 16?
(1)
All of a, b, c and d are positive with a
(2)
a+b+c+d=8
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b
c
d.
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Number Properties Guide – DS Questions
113
Solve yourself:
160.
What is the value of a ⇥ b ⇥ c ⇥ d, if a, b, c and d are positive integers?
(1)
(2)
a+b+c+d=9
a
b>c>d
Solve yourself:
161.
105
Is abc
an integer, given that a, b and c are integers?
(1)
abc is prime.
(2)
1  a  5, 1  b  7 and 3  c  10
Solve yourself:
162.
If Pn = a1 ⇥ a2 ⇥ a3 . . . an , where a1 > 0 and a2 < 0, is P8 > 0?
(1)
All terms having an odd suffix have the same value.
(2)
All terms having an even suffix have the same value.
Solve yourself:
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114
163.
Number Properties Guide – DS Questions
If m and n are positive numbers, is m
(1)
(2)
n?
m3
1
n
p
3
m
1
n
Solve yourself:
164.
Is (a + b + c + d + e) even, where a, b, c, d and e are positive integers?
(1)
(a ⇥ b ⇥ c ⇥ d ⇥ e) is odd.
(2)
(a ⇥ b ⇥ c ⇥ d) is odd.
Solve yourself:
165.
Is (a
(1)
(2)
b)(b
c) even, where a, b and c are positive integers with a > b > c?
(a ⇥ b ⇥ c) is odd.
✓
◆
a⇥b
is even.
c
Solve yourself:
166.
What is the value of q, if 3p + q2 = 42?; where p and q are positive integers.
(1)
p is a prime number
(2)
p can be expressed as the sum of two positive integers in only one way
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Number Properties Guide – DS Questions
115
Solve yourself:
167.
If a, b and c are prime numbers, is c = 2?
(1)
(2)
a + b + c = 56
a>b>c
Solve yourself:
168.
Is (n
1)(n
3) even?
(1)
(n2
(2)
(n + 1)(n + 5) is odd.
9) is odd.
Solve yourself:
169.
What is the value of the positive integer p?
(1)
(2)
p
22 = 256
2
2p = 512
Solve yourself:
170.
If ki = p ⇥ i + i for all integer values of i, what is the value of p?
(1)
k10 = 2k9
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116
Number Properties Guide – DS Questions
(2)
k7 + k8 = 0
Solve yourself:
171.
If p and q are positive integers, is p a factor of q?
(1)
(2)
2p 1 = p 2 with p < 4
3q3
6 = 75
Solve yourself:
172.
If n is a positive integer, does n have divisors other than 1 and n?
(1)
n<6
(2)
5 < 4n < 13
Solve yourself:
173.
Is p a positive even number?
(1)
If p is divided by 5, the remainder is 2.
(2)
If p is divided by 8, the remainder is 2.
Solve yourself:
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Number Properties Guide – DS Questions
174.
117
Is u2 > 64?
(1)
u>
(2)
u<8
8
Solve yourself:
175.
Is 0 < p < 1?
p
(1) 0 < 3 p < 1
1
(2) p 4 =
16
Solve yourself:
176.
What is the greatest common divisor of the positive integers m and n?
(1)
(2)
m
n
The greatest common divisor of the positive integers 4 and 4 is 12.
m
n
2 and 2 are even.
Solve yourself:
177.
If x > 0, what fraction of y is x?
(1)
(2)
x2
2xy 3y 2 = 0
x
9y
x% of
=
y
100
Solve yourself:
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118
178.
Number Properties Guide – DS Questions
If 375
5n is an integer, how many values n can assume?
(1)
n>5
(2)
n has greater than two factors.
Solve yourself:
179.
Is
⇣
240+150n
n
⌘
an integer?; where n 6= 0.
(1)
n is a factor of 75.
(2)
n is a factor of 30.
Solve yourself:
180.
If n is a prime number, what is the remainder when n2 is divided by 6?
(1)
n>2
(2)
n>7
Solve yourself:
181.
What is the value of the non-zero number p?
(1)
(2)
|p|3 + p 3 = 0
|p|2 =
2p
Solve yourself:
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Number Properties Guide – DS Questions
182.
119
Which one among |a|, |b|, |c| and |d| is maximum, if a < b < c < d?
(1)
(2)
a=
d=
2d
b
Solve yourself:
183.
If x, y and z are positive integers, what is the value of (|x
(1)
(2)
x + z = 2y
y| + |y
z|)?
xz = y 2
Solve yourself:
184.
Is
(1)
(2)
x
an integer?; where x and y are positive integers.
y
y(2x + 1) is an even integer.
x = (2x
1)2
Solve yourself:
185.
Is xy divisible by 60?
(1)
x > 3 and is a factor of 12.
(2)
y is an even multiple of 15.
Solve yourself:
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120
186.
Number Properties Guide – DS Questions
(1)
⇣
(2)
The GCD (Greatest Common Divisor) of x and y is 1.
Can
1
1
x + y
⌘
be expressed as z1 ?; where x, y and z are distinct positive integers
x and y are both less than 7.
Solve yourself:
187.
What is the value of abc?
(1)
a, b and c are distinct prime numbers less than 10.
(2)
ab = 15, bc = 21 and ac = 35.
Solve yourself:
188.
If [x] denotes the least integer greater than or equal to x, is [x] < 0?
(1)
(2)
8x 2 + 6x + 1 = 0
2x 2
x
Solve yourself:
189.
1=0
Each of the three positive numbers is squared and divided by 3. What is the sum of the
remainders obtained upon their division?
(1)
The numbers are consecutive integers.
(2)
The numbers lie between 10 and 30.
Solve yourself:
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Number Properties Guide – DS Questions
190.
121
What is the sum of the digits of a 3-digit number?
(1)
The product of the digits is 36.
(2)
All the three digits are not distinct and none of the digits is 4.
Solve yourself:
191.
What is the remainder when the number 2N is divided by 10?
(1)
The number N when divided by 100 leaves a remainder 24.
(2)
The number N when divided by 5 leaves a remainder 4.
Solve yourself:
192.
If k and r are integers, what is the value of k?
p
(1)
16 ⇥ 10r = k ⇥ 103
(2)
100 < 2.36 ⇥ 10k < 1, 000
Solve yourself:
193.
For positive integers x, y, and n, is (x n+1
x > y.
(1)
n is even
(2)
n is a multiple of 3
xy n ) divisible by (x 2
y 2 )? Given that
Solve yourself:
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122
194.
Number Properties Guide – DS Questions
If a is a positive integer, is (a3 + a) divisible by 6?
(1)
(2)
For any positive integer n, [n(n + 1)(n + 2)] is divisible by a!.
a is an odd integer.
Solve yourself:
195.
If p is a positive integer, how many integer values of n are possible if
p
45 + p?
(1)
p<6
(2)
4 < p < 10
p
45 < n <
Solve yourself:
196.
If c < 0, is
(1)
(2)
c
> 0?
a
b3
>0
|a|
|b|
>0
a+c
Solve yourself:
197.
✓
◆
1
If x is a positive number, what is the value of x +
?
x
✓
◆
1
(1)
x2 + 2 = 2
x
✓
◆
p
1
(2)
x+p
=2
x
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Number Properties Guide – PS Questions
123
Solve yourself:
198.
If P =
(1)
(2)
⇣
⌘
1
1 2
+
a
2b
, is P > 1?
4
ab
b
2
a=4
a=
Solve yourself:
199.
The symbol ‘r’ represents a mathematical operation between two numbers a and b such
that a r b = (ab + ba ). What is the value of x if x r 2 = 17?
(1)
x is a positive integer.
(2)
x is a prime number.
Solve yourself:
200.
If a and b are distinct integers, is (2a)b > 30?
(1)
a and b are single digit primes.
(2)
a+b =5
Solve yourself:
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128
Number Properties Guide – Answer Key
4.3 Assorted Questions
4.3.1
Problem Solving
(101) D
(114) B
(127) E
(102) E
(115) D
(128) D
(103) B
(116) C
(104) A
(117) C
(105) B
(118) C
(106) D
(119) B
(131) C
(107) C
(120) C
(132) B
(108) D
(121) D
(133) A
(109) C
(122) D
(110) A
(123) D
(111) C
(124) B
(112) C
(125) A
(136) E
(113) A
(126) B
(137) D
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(129) C
(130) D
(134) A
(135) C
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