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89 arithmetic/tests.for.divisibility/eleven.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

89 arithmetic/tests.for.divisibility/eleven.p


What is the test to see if a number is divisible by eleven?

arithmetic/tests.for.divisibility/eleven.s

If the alternating sum of the digits is divisible by eleven, so is the number.

For example, 1639 leads to 9 - 3 + 6 - 1 = 11, so 1639 is divisible by 11.

Proof:
Every integer n can be expressed as
n = a1*(10^k) + a2*(10^k-1)+ .....+ a_k+1
where a1, a2, a3, ...a_k+1 are integers between 0 and 9.
10 is congruent to -1 mod(11).
Thus if (-1^k)*a1 + (-1^k-1)*a2 + ...+ (a_k+1) is congruent to 0mod(11) then
n is divisible by 11.

 

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