NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 1: Number System - Fill in the Blanks

Question 148

Divisibility rule for 11 — step by step

1. Write the number and mark its places from the right.
   1st place = odd, 2nd place = even, 3rd place = odd, 4th place = even, and so on.
2. Add the digits in the odd places (1st, 3rd, 5th, ...).
   Let this sum be \(S_{\text{odd}}\).
3. Add the digits in the even places (2nd, 4th, 6th, ...).
   Let this sum be \(S_{\text{even}}\).
4. Find the difference:
   \(D = \lvert S_{\text{odd}} - S_{\text{even}} \rvert\).
5. Check \(D\):
   If \(D = 0\) or \(D\) is a multiple of 11 (that is, \(11, 22, 33, \dots\)),
   then the number is divisible by 11.

Quick example: 3080 (digits from right: 0, 8, 0, 3)

Odd places: 1st = 0, 3rd = 8 → \(S_{\text{odd}} = 0 + 8 = 8\)
Even places: 2nd = 0, 4th = 3 → \(S_{\text{even}} = 0 + 3 = 3\)
Difference: \(D = \lvert 8 - 3 \rvert = 5\)

Here \(D=5\) is not a multiple of 11, so 3080 is not divisible by 11.

Key idea: “Difference of (odd-place sum) and (even-place sum) is 0 or a multiple of 11.”