Let N = am 10m + ⋯ + a2 102 + a1 10 + a0, where 0 ≤ ak ≤ 9, be the decimal expansion of a...

Let N = am 10m + ⋯ + a2 102 + a1 10 + a0, where 0 ≤ ak ≤ 9, be the decimal expansion of a positive integer N.

(a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer

[Hint: If n is even, then 103n ≡ 1, 103n + 1 ≡ 10, 103n + 2 ≡ 100 (mod 1001); if n is odd, then 103n ≡ −1, 103n + 1 ≡ −10, 103n + 2 ≡ −100 (mod 1001).]

(b) Prove that 6 divides N if and only if 6 divides the integer