Math Help - prove that then n is divisible by 11

1. prove that then n is divisible by 11

Prove that is n is a positive integer such that the alternately adding and subtracting its digits in decimal representation yeilds divisible by 11 number, then n is divisible by 11.

2. Originally Posted by mandy123
Prove that is n is a positive integer such that the alternately adding and subtracting its digits in decimal representation yeilds divisible by 11 number, then n is divisible by 11.
$n=a_0+a_1~10+a_2~10^2+...+a_k~10^k$

Now $10^r \equiv (-1)^r \mod 11$

so:

$n \equiv a_0 - a_1 + a_2 +\ ...\ +(-1)^ka_k \mod 11$

That is $n$ is congurent to the number obtained by alternately adding and subtracting its digits in decimal representation, but $n$ is congruent to $0$ modulo $11$, and therefore so is alternately adding and subtracting its digits in decimal representation.

RonL