Divisibility Rules -
Solved Problems
To completely understand this chapter, go through
Divisibility
Rules - Basic Concept
Problems on Divisibility by 3 :
1). Find the smallest value of
the variable of x for which the number 187x25 is divisible by 3.
a)0 b)1 c)2 d)4
Answer: B
Explanation:
To check divisibility by 3, we
have to compute the sum of the digits:
1+8+7+x+2+5 = 23+x
For divisibility by 3, the sum
of the digits should also be divisible by 3.
So, 23+x should be divisible by
3.
The least value for x is 1,
making 23+x=24 divisible by 3.
Solved Problems on Divisibility
Rule of 3:
2). Find the largest positive
integer value of P for which 65x44 is divisible by 3.
a)0 b)2 c)5 d)8
Answer : D
Explanation:
Check divisibility by 3 by
adding the digits: 6+5 + x+4 +4 = 19+x
For divisibility by 3, the sum
should also be divisible by 3.
So, 19+x should be divisible by
3.
The largest positive integer
value for x is 8, making 19+8=27 divisible by 3.
Solved Problems on Divisibility
by 4:
3). What is the least value of
the variable x for which the number 264x2 is divisible by 4?
a)1 b)2 c)4 d)5
Answer: B
Explanation:
For a number to be divisible by
4, the number formed by the last two digits of the number must be divisible by
4.
So, 2x should be divisible by 4.
The least value for x is 2,
making 2x=24 divisible by 4.
Problems on Divisibility Test –
Divisibility Test by 4
4). What is the largest value of
the variable x for which the number 2653x4 is divisible by 4?
a)2 b)3 c)7 d)8
Answer : D
Explanation :
For a number to be divisible by
4, the number formed by the last two
digits of the number must be divisible by 4.
So, x4 should be divisible by 4.
The largest value for x is 8,
making x4=84 divisible by 4.
Divisibility Test for 6: Solved Problems on Divisibility by 6
5). 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: The number should 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+*
22+* must be divisible by 3= > 22+2 =24
and 22 +5=7.
2 and 7 satisfy the condition ii.
The missing digit must satisfy the condition (i)
also
Out of 2 and 5, only 2 is even.
Checking the divisibility by 6: Solved Example
6) If 522x is a three-digit number with a digit x. If
the number is a multiple of 6, what is the value of the digit x?
a) 1
b) 2
c) 3
d)
4
e)6
Answer: E
Explanation:
If a number is divisible by 6, it is a multiple
of 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 of the given options
552x is divisible by 3 If the 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 by 8: Questions on Divisibility Rules
7) Find the least 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:
If a number is exactly divisible by 8, then the number formed by
the last 3 digits of the number must be divisible by 8.
Here the last 3 digits are 8x4.
Put each value in the given options in the place of x and
check it.
Option b, 824 which is exactly divisible by 8.
So the answer is option b.
Solved Problems on Divisibility Rules – Divisibility by 8
8) What is the smallest number that should be added to 3681563 to
make it exactly divisible by 8?
a) 2 b) 3 c) 4 d) 5
Answer: D
Explanation:
If a number is divisible by 8, the number formed by its last three
digits must be divisible by 8.
Here, the last three digits of 3681563 are 563, and the next
multiple of 8 is 568.
To make it divisible by 8, the smallest number to add is 5.
So, 5 must be added to 81563 to make it divisible by 8.
Solved Problem on Divisibility Rules for 8
9) A number is formed by
writing the first 73 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
Correct
Option: D
Explanation:
If the number formed by the last three digits of the number is exactly divisible
by 8, then the number is exactly 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 the first 67 natural numbers is 273.
Therefore, the remainder when 273 is
divisible by 8 is 1.
Divisibility Rules Solved Problems for Bank Tests - Divisibility by 9
10) Find the least value of * for which 3* 8712 is divisible by
9
a)5 b)6 c)8 d)12
Answer:
B
Explanation
:
Solution: Let the required value be x.
Then, (3 + x + 8 + 7 + 1 + 2) = (21 + x) should be divisible by
9.
⇒ x = 6
Divisibility Rules Solved Problem – Divisibility by 9.
11) 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
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 Tests: Examples of divisibility by 9
12) 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 digits of the number 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 by 9: Solved Examples
13) Find the least 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.