This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
What is the test to see if a number is divisible by eleven?
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.
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.