State and prove the test of divisibility of a positive integer $$a$$ by 11.

PLACEHOLDER ANSWER - this is a proof question. The test is that $$a$$ is divisible by 11 exactly when the alternating sum of its digits, units minus tens plus hundreds and so on, is divisible by 11. If $$a = d_0 + 10d_1 + 10^2 d_2 + \cdots + 10^n d_n$$, then since $$10 \equiv -1 \pmod{11}$$, $$10^k \equiv (-1)^k$$, so $$a \equiv d_0 - d_1 + d_2 - \cdots + (-1)^n d_n \pmod{11}$$. Hence 11 divides $$a$$ if and only if it divides this alternating sum.