Prove the following statements about the positive integer n with decimal expansion

Question

(a)n ≡ Σ ak (mod 3) (view the image)

(b) n ≡ a1a0 (mod 4)

https://i.stack.imgur.com/N4Ogw.png

Answer 1

For the first exercise, note that if the decimal expansion of n = $a_ma_{m-1}\ldots a_1a_0$, then $$ n = \sum_{k=0}^ma_k10^k $$ Also recall that 1 $\equiv 10^k$(mod 3) for every k. So $$n (mod 3) \equiv \sum_{k=0}^ma_k10^k(mod 3) \equiv\sum_{k=0}^ma_k(mod 3).\checkmark $$

Similarly for the second exercise note that 0 $\equiv 10^k$ for k $\ge$2. So $$n(mod 4) \equiv \sum_{k=0}^ma_k10^k(mod 4)\equiv\sum_{k=0}^1a_k10^k(mod 4) \equiv a_1a_0(mod4). \checkmark $$