(a) Prove that for all integers n 1, 10n (1) n mod11.

(b) Use part (a) to prove that a positive integer is divisible by 11 if, and only if, the alternating sum of its digits is divisible by 11. (For instance, the alternating sum of the digits of 82,379 is 8 2 + 3 7 + 9 = 11 and 82,379 = 11 7489)