(Solved): Q2. Let \( x \in \mathbb{N} \) and let \( d \) be the units digit of \( x \). (a) Prove that \( \fr ...

Q2. Let \( x \in \mathbb{N} \) and let \( d \) be the units digit of \( x \). (a) Prove that \( \frac{x-d}{10} \in \mathbb{N} \cup\{0\} \). (b) Prove that \( \frac{x-d}{10}+7 d \equiv 0(\bmod 23) \) if and only if \( x \equiv 0(\bmod 23) \). (This result can be use to recursively test whether a natural number is divisible by 23.)