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.
$\displaystyle n=a_0+a_1~10+a_2~10^2+...+a_k~10^k$
Now $\displaystyle 10^r \equiv (-1)^r \mod 11$
so:
$\displaystyle n \equiv a_0 - a_1 + a_2 +\ ...\ +(-1)^ka_k \mod 11 $
That is $\displaystyle n$ is congurent to the number obtained by alternately adding and subtracting its digits in decimal representation, but $\displaystyle n$ is congruent to $\displaystyle 0$ modulo $\displaystyle 11$, and therefore so is alternately adding and subtracting its digits in decimal representation.
RonL