## If N5921 is divisible by 11, find the value of the smallest natural number N.

**यदि N5921, 11 से विभाज्य है, तो सबसे छोटी प्राकृतिक संख्या N का मान ज्ञात कीजिए।**

a) 6

b) 9

c) 7

d) 8

**Answer : d) 8**

Uttar Pradesh Police SI Exam paper – Mathematics

### Solution :

### Divisibility rule for 11.

This rule states that a number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.

Let’s apply this rule to ( N5921 ):

**Identify the positions of the digits:**- Odd positions: ( N, 9, 1 )
- Even positions: ( 5, 2 )

**Calculate the sums:**- Sum of digits in odd positions: ( N + 9 + 1 = N + 10 )
- Sum of digits in even positions: ( 5 + 2 = 7 )

**Find the difference:**- Difference = ( (N + 10) – 7 = N + 3 )

For ( N5921 ) to be divisible by 11, ( N + 3 ) must be either 0 or a multiple of 11. The smallest natural number ( N ) that satisfies this condition is when ( N + 3 = 11 ).

**Solve for ( N ):**- ( N + 3 = 11 )
- ( N = 8 )

Therefore, the smallest natural number ( N ) that makes ( N5921 ) divisible by 11 is 8.