What is the rule for 11, and why the alternating sum?

Add and subtract the digits alternately; if that result is 0 or a multiple of 11, so is the number — because 10 ≡ −1 (mod 11).

Since 10 ≡ −1 (mod 11), consecutive place values flip sign (+1, −1, +1, …), turning the number into an alternating digit sum mod 11:

• 918,082 → 9 − 1 + 8 − 0 + 8 − 2 = 22 = 2 × 11 → divisible. ✓
• 2,728 → 2 − 7 + 2 − 8 = −11 → divisible. ✓ (2728 ÷ 11 = 248)

Starting the alternation from either end gives an equivalent result mod 11.