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Aptitude Tests - Divisibility Rules - Solved Problems To completely understand this chapter, go through   Divisibilit...

# Aptitude Tests

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## Divisibility Rules - Solved Problems

To completely understand this chapter, go through DivisibilityRules - Basic Concept.

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
Explanation:
If a number is divisible by 9, then 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
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 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 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
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 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.

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
Explanation:

In aptitude tests, we get questions on divisibility by 11.
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
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 value of ‘x’ so that the number 73818x4 is divisible by 8
a)1          b) 2            c) 3              d) 4         e)6
Explanation:
A number is exactly divisible by 8, then the last 3 digits of the numbers must be divided 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 the answer is option b.

Divisibility Rules Solved Problem 7.
Find the least of the value of ‘*’ so that the number 37*124 is divisible by 9?
a)1            b) 10          c) 2            d) 3             e)4
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
Explanation:
For a number to be divisible by 88, the number must be divisible by 8 and 11.
Write  88 as a 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.

Divisibility Rules Solved Problem 9.
What is the smallest number which must be added to 8261955 so as to obtain a sum which is divisible by 11?
a) 1            b)2                 c)3            d)4           e)5
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
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 question is a difficult question on divisibility rules.
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.

For more solved problems and online tests on quantitative aptitude, visit http://www.9exams.com