In this question, well figure out how to find the greatest common divisor of three integers.

(a) Prove: for any positive integers a, b, d, if d | a and d | b, then d | GCD(a, b).

(b) Given positive integers a, b, c, we define GCD(a, b, c) to be the positive integer d such that: i. d | a and d | b and d | c ii. for any positive integer e, if e | a and e | b and e | c, then e d. Prove: GCD(a, b, c) = GCD(a, GCD(b, c)). (Hint: you might find question 6a useful here)

(c) Use question 6b to help you calculate GCD(272715979, 347551889, 247214621).

(d) For all integers a, b, c, e, prove that the Diophantine equation ax + by + cz = e has a solution if and only if GCD(a, b, c) | e.