Alternating Sum Uncovered

A number is divisible by 11 if the alternating sum of its digits equals 0 or a multiple of 11. For 918,082: calculate 9 minus 1 plus 8 minus 0 plus 8 minus 2, which equals 22 — a multiple of 11, confirming divisibility. This elegant test works for numbers of any length and requires nothing more than addition and subtraction.

This test comes from the fact that powers of 10 alternate between leaving remainder 1 and remainder -1 when divided by 11. The pattern — 10¹ mod 11 = -1, 10² mod 11 = 1, 10³ mod 11 = -1 — creates the alternating sum rule. Modular arithmetic makes divisibility elegant.