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** Aptitude Tests**

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**
Divisibility Rules -
Solved Problems**

**Divisibility Rules Solved Problem 1**.

What is the smallest number that should be added to 27452
to make it exactly divisible by 9?

a)1 b)2 c)7 d)8 e)9

**Answer : C**

**Explanation:**

If a number is divisible by 9, the
sum of its digits must be a multiple of 9.

Here, 2+7+4+5+2=20, the next multiple
of 9 is 27.

7 must be added to

**27452**to make it divisible by 9.**Divisibility Rules Solved Problem 2**

If 522x is a three digit number with as a digit x .
If the number is divisible by 6, What is the value of the digit x is?

a) 1 b) 2 c) 3 d)
4 e)6

**Answer: E**

**Explanation:**

If a number is divisible by 6 , it must be
divisible by both 2 and 3

In 522x, to this number be divisible
by 2, the value of x must be even. So it
can be 2,4 or 6 from given options

552x is divisible by 3, If sum of its
digits is a multiple of 3.

5+5+2+x =12+x
,

If put x =2 , 12+2=14 not a multiple of 3

If put x =4 , 12+6=18 is a multiple of 3

If put x =6 , 12+2=14 not a multiple of 3

The value
of x is 6.

**Divisibility Rules- Solved Problem 3**.

What is the least value of x such that 7648x is divisible by 11?

a) 1 b)
2 c) 3 d)
4 e)5

**Answer : A**

**Explanation:**

A number is divisible by 11, when difference between the sum
of digits at even places and at odd places is 0 or multiple of 11

The given number is 7648x.

(Sum of digits at EVEN places) – (sum of digits at ODD
places)=0

(6
+ 8 ) - ( X + 7 + 6 ) = 0

14
- ( X + 13 ) = 0

Here the value of x must be 1 .

So , the least value of x is 1 .

So , the least value of x is 1 .

**Divisibility Rules Solved Problem 4**

If M183 is divisible by 11, find the value of the smallest
natural number M ?

a)
5 b)
6 c)
7 d)
9 e)8

**Answer : C**

**Explanation:**

A number is divisible by 11, when the difference between the sum of digits at even places and at odd places is 0 or multiple of 11

The given number is M183.

(Sum of digits at EVEN places) – (sum of digits at ODD
places)=0

(8
+ M )- (3+1)= 0

(8
+ M ) - 4 = 0

Here the value of M must be 7 .

**Divisibility Rules Solved Problem 5**

What least whole number should be added to 532869 to make it
divisible by 9?

a)2 b)1 c)3 d)4 e)5

**Answer : C**

**Explanation**:

If a number is divisible by 9, the sum of its digits must be a
multiple of 9.

Here, 5+3+2+8+6+9=33,
the next multiple of 9 is 36.

3 must be added to

**532869**to make it divisible by 9.**Divisibility Rules Solved Problem 6.**

Find the least of value of ‘x’ so that the number 73818x4 is
divisible by 8

a)1 b)
2 c)
3 d)
4 e)6

**Answer : B**

**Explanation:**

A number is exactly divisible by 8, then the last 3 digits of the
numbers must be divisible by 83

Here the last 3 digits are 8x4.

Put each value in given options in the place of x and check whether it is divisible by 8 or not .

Option b , 824 which is exactly
divisible by 8.

So answer is option b.

**Divisibility Rules Solved Problem 7.**

Find the least of value of ‘*’ so that the number 37*124 is
divisible by 9 ?

a)1 b)
10 c)
2 d) 3 e)4

**Answer : A**

**Explanation:**

if a number is divisible by 9, the sum of its digits must be a
multiple of 9.

Here, 3+7+*+1+2+4=17+*

Here the value of * must be 1 because the next multiple of 9 is
18.

**Divisibility Rules Solved Problem 8.**

For a number to be divisible by 88 it should be:

a) Divisible by 8.

b) Divisible by 8 and 22.

c) Divisible by 8 and 11.

d) Divisible by 4,22

e) can’t say

**Answer: C**

**Explanation:**

For a number to be divisible by 88, the number must be
divisible by 8 and 11.

Write 88 as product of two factors : 22
,2

11
,8

44,2

Of these pairs , 11 and 8 are co primes.

So the number must be divisible by 8 and 11.

So the number must be divisible by 8 and 11.

**Divisibility Rules Solved Problem 9.**
a) 1 b)2 c)3 d)4 e)5

**Answer : B**

**Explanation:**

For divisibility by 11, the difference of sums of digits
at even and odd places must be either zero or divisible by 11.

For 8261955, Difference =(8+6+9+5) -(2+1+5)=28-8=20.

The units digit is at odd place. So we add 2 to the number

=> 8261955 +2 = 8261957

Now , (8+6+9+7) -(2+1+5)=30-8=22 => 22 is a multiple of
11 and hence 8261957 is also divisible by 11.

**Divisibility Rules Solved Problem 10.**

What is the missing digit which makes the number 9724* exactly
divisible by 6 ?

a)2 b)4 c)5 d)6 e)8

**Answer : A**

**Explanation:**

Divisibility by 6 requires that the number be
divisible by 2 as well as 3 , i.e, the following 2 conditions must be met

i) Unit digit be Zero or even

ii) Sum of digits be divisible by 3

The given number is 9724*

Sum of the digits =9 +7 +2 +4 +* =22+*

The digit which on being added to 22 will
give the sum divisible by 3 are

22+2 =24 and 22 +5=7.

2 and 7 satisfy the condition ii.

The blank space is at unit's place. So the
missing digit must satisfy the condition (i) also

Out of 2 and 5, only 2 is even.

**Problem on Divisibility Rules for 8 Question 11**

A
number is formed by writing first 67 natural numbers in front of each other as
1234567891011....What is the remainder when this number is divided by 8?

a)7 b)6 c)3 d) 1 e)0

**Correct Option: c**

**Explanation:**

This the next level difficulty question for aptitude tests.

If the number formed by the last three digits of the number is divisible by 8,
then the number is divisible by 8.

We apply the same
rule to find the remainder when a number is divided by 8.

Now the last 3
digits in the number formed by first 67 natural numbers is 667.

Therefore, the
remainder when 667 is divisible by 8 is 3.

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