25. On dividing a number by 56, we get 29 as remainder. On dividing the
same number by 8, what will be the remainder ?
A. 2
B. 3
C. 4
D. 5
Solution 1
------------------------------------------------------------------------------------Number = 56x + 29 (∵ since the number gives 29 as remainder on dividing by 56)
= (7 × 8 × x) + (3 × 8) + 5
Hence, if the number is divided by 8, we will get 5 as remainder.
------------------------------------------------------------------------------------Solution 2
------------------------------------------------------------------------------------Number = 56x + 29
Let x = 1. Then the number = 56 × 1 + 29 = 85
85 ÷ 8 = 10, remainder = 5
28. How many 3 digit numbers are completely divisible 6 ?
A. 146
B. 148
C. 150
D. 152
Explanation :
100/
6
= 16, remainder = 4. Hence 2 more should be added to 100 to get the
minimum
3 digit number divisible by 6.
=> Minimum 3 digit number divisible by 6 = 100 + 2 = 102
999/
6
= 166, remainder = 3. Hence 3 should be decreased from 999 to get
the maximum
3 digit number divisible by 6.
=> Maximum 3 digit number divisible by 6 = 999 - 3 = 996
Hence, the 3 digit numbers divisible by 6 are 102, 108, 114,... 996
This is Arithmetic Progression with a = 102 ,d = 6, l=996
Number of terms =(l−a)/d+1=(996−102)/6+1=(894/6)+1=149+1=150
31. If x and y are positive integers such that (3x + 7y) is a multiple of 11,
then which of the followings are divisible by 11?
A. 9x + 4y
B. x + y + 4
C. 4x - 9y
D. 4x + 6y
Explanation :
By hit and trial method, we get x=5 and y=1 such that 3x + 7y = 15 + 7 =
22 is a multiple of 11.
Then
(4x + 6y) = (4 × 5 + 6 × 1) = 26 which is not divisible by 11
(x + y + 4) = (5 + 1 + 4) = 10 which is not divisible by 11
(9x + 4y) = (9 × 5 + 4 × 1) = 49 which is not divisible by 11
(4x - 9y) = (4 × 5 - 9 × 1) = 20 - 9 = 11 which is divisible by 11
2nd---3x + 7y) isa multiple of 11
Now we need to find out a value for x and a value for y which satisfies the
equation
Now we try randomly
For example, Take y = 1. (the minimum value for y)
Then 3x + 7 should be a multiple of 11
for x = 1, 3x + 7= 3 + 7 = 10
for x = 2, 3x + 7= 3*2 + 7 = 13
for x = 3, 3x + 7= 3*3 + 7 = 16
for x = 4, 3x + 7= 3*4 + 7 = 19
for x = 5, 3x + 7= 3*5 + 7 = 22.
So, when y = 1 and x = 5 satisfies this expression
76. On dividing 2272 as well as 875 by 3-digit number N, we get the same
remainder. What is the sum of the digits of N?
A. 11
B. 10
C. 9
D. 8
Explanation :
Let 2272 ÷ N = a, remainder = r
=> 2272 = Na + r ----------------------------(Equation 1)
Let 875 ÷ N = b, remainder = r
=> 875 = Nb + r ----------------------------(Equation 1)
(Equation 1) - (Equation 2)
=> 2272 - 875 = [Na + r] - [Nb + r] = NA - Nb = N(a - b)
=> 1397 = N(a - b) ----------------------------(Equation 3)
It means 1397 is divisible by N
But 1397 = 11 × 127
[Reference1 :how to find factors of a number?]
[Reference2 :Prime Factorization?]
You can see that 127 is the only 3 digit number which perfectly divides
1397
=> N = 127
sum of the digits of N = 1 + 2 + 7 = 10
86. What is the digit in the unit place of the number represented by (7 95 358)
A. 4
B. 3
C. 2
D. 1
Explanation :
First let's find out the unit digit of 795
795 = [(74)23 × 73]
Hence, Unit Digit of 795
= Unit Digit of (74)23 × Unit Digit of 73 ----(Equation 1)
Unit Digit of [(74)23]
= Unit Digit of [(7×7×7×7)23]
= Unit Digit of [(9 × 9)23] (∵ 7×7=49 and 9 is the unit digit of 49)
= Unit Digit of [123] (∵ 9×9=81 and 1 is the unit digit of 81)
= 1 ----(Equation 2)
Unit Digit of 73
= Unit Digit of [7 × 7 × 7]
= Unit Digit of [9 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)
= 3(∵ 9 × 7 = 63 and 3 is the unit digit of 63) ----(Equation 3)
Hence from Equation 1,Equation 2 and Equation 3,
unit digit of 795 = 1 × 3 = 3 ----(Equation A)
Similarly we can find out unit digit of 358
358 = [(34)14 × 32]
Hence, Unit Digit of 358
= Unit Digit of (34)14 × Unit Digit of 32 ----(Equation 4)
Unit Digit of [(34)14]
= Unit Digit of [(3×3×3×3)14]
= Unit Digit of [(9 × 9)14] (∵ 3×3=9)
= Unit Digit of [114] (∵ 9×9=81 and 1 is the unit digit of 81)
= 1 ----(Equation 5)
Unit Digit of 32 = 9----(Equation 6)
Hence from Equation 4,Equation 5 and Equation 6,
Unit Digit of 358 = 1 × 9 = 9 ----(Equation B)
We have already found out that
Unit Digit of 795 = 3 (From equation A)
and Unit Digit of 358 = 9 (From equation B)
Hence, Unit Digit of (795 - 358)
= Unit Digit of 795 - Unit Digit of 358
= unit digit of [larger number of last digit 3 - smaller number of last digit 9
](∵ 795 > 358)
= 4 (∵ 4 is the unit digit when a smaller number of
last digit 9 is subtracted from a larger number of last digit 3. example : 113
- 19 = 94)
87. What is the unit digit in the number represented by [365 × 659 × 771]
A. 1
B. 2
C. 3
D. 4
nswer : Option D
Explanation :
First let's find out the unit digit of 365
365 = [(34)16 × 3]
Hence, Unit Digit of 365
= Unit Digit of (34)16 × 3
= Unit Digit of [(3×3×3×3)16] × 3
= Unit Digit of [(9 × 9)16] × 3 (∵ 3×3=9)
= Unit Digit of [116] × 3 (∵ 9×9=81 and 1 is the unit digit of 81)
=1×3
= 3 ----(Equation A)
Unit digit of 659 = 6 (∵ 6×6=36 and unit digit remains as 6 always) ---(Equation B)
771 = [(74)17 × 73]
Hence, Unit Digit of 771
= Unit Digit of (74)17 × Unit Digit of 73 ----(Equation 1)
Unit Digit of [(74)17]
= Unit Digit of [(7×7×7×7)17]
= Unit Digit of [(9 × 9)17] (∵ 7×7=49 and 9 is the unit digit of 49)
= Unit Digit of [117] (∵ 9×9=81 and 1 is the unit digit of 81)
= 1 ----(Equation 2)
Unit Digit of 73
= Unit Digit of [7 × 7 × 7]
= Unit Digit of [9 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)
= 3(∵ 9 × 7 = 63 and 3 is the unit digit of 63) ----(Equation 3)
Hence from Equation 1,Equation 2 and Equation 3,
unit digit of 771 = 1 × 3 = 3 ----(Equation C)
We have already found out that
Unit Digit of 365 = 3 (From equation A)
Unit Digit of 659 = 6 (From equation B)
Unit Digit of 771 = 3 (From equation C)
Hence, unit digit in the number represented by [365 × 659 × 771]
= unit digit of [3 × 6 × 3]
= unit digit of [8 × 3](∵ 3 × 6 = 18 and 8 is the unit digit of 18)
= 4 (∵ 8 × 3 = 24 and 4 is the unit digit of 24)
92. A three-digit number 4a3 is added to another three-digit number 984 to
give a four digit number 13b7, which is divisible by 11. What is the value of
(a + b)?
A. 9
B. 10
C. 11
D. 12
Answer : Option B
Explanation :
(Reference : Divisibility by 11 rule)
4a3
984
---------13 b 7
=> a + 8 = b -----------(Equation 1)
13b7 is divisible by 11
=> (1 + b) - (3 + 7) is 0 or divisible by 11
=> (b - 9) is 0 or divisible by 11 -----------(Equation 2)
Assume that (b - 9) = 0
=> b = 9
Substituting the value of b in Equation 1,
a+8=b
a+8=9
=> a = 9 - 8 = 1
If a = 1 and b= 9,
(a + b) = 1 + 9 = 10
10 is there in the given choices. Hence this is the answer.
8. A number when divided by 75 leaves 34 as remainder. What will be the
remainder if the same number is divided by 65?
A. 3
B. 1
C. 6
D. 9
Answer : Option D
Explanation :
Let the number be x
Let x ÷ 75 = p and remainder = 34
=> x = 75p + 34
= (25p × 3) + 25 + 9
= 25(3p + 1) + 9
Hence, if the number is divided by 25, we will get 9 as remainder