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 ):

1. Identify the positions of the digits:
   • Odd positions: ( N, 9, 1 )
   • Even positions: ( 5, 2 )
2. Calculate the sums:
   • Sum of digits in odd positions: ( N + 9 + 1 = N + 10 )
   • Sum of digits in even positions: ( 5 + 2 = 7 )
3. 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 ).

4. Solve for ( N ):
   • ( N + 3 = 11 )
   • ( N = 8 )

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