Please help me with writing proofs using complete sentences(only words) for each question.

8. You probably learned about the Rule of 3 when you learned how to multiply and factor natural numbers. The Rule of 3 states that a natural number n is divisible by 3 if and only if the sum of its digits is divisible by 3. In terms of congruences, this says: n E 0 (mod 3) 4:} the sum of the digits of n is divisible by 3. Prove the Rule of 3. Hint: ifa natural number n has digits (10,411, . . . , ak then n = a0 + 100,1 + . . . + 10kak. 9. Prove that for all integers a, b which are not both 0, and for all integers d, d]gcd(a, b) if and only if d|a and dlb. Hints. a. Recall Bezout's theorem: for all integers m and n which are not both 0, there exist integers a: and y such that mm + ny = gcd(m, n). b. Recall that divisibility is transitive: if sit and tin then s Iv