5. This final question will walk us through the solutions (or lack thereof) to the linear diophantine equations 3n + 6m = 1 and 3n + 5m = 1.

(a) Prove the following theorem:

Theorem. There do not exist integer solutions m, n Z such that 3n + 6m = 1.

Strategy:

Proof:

(b) Now, consider the linear diophantine equation 3n + 5m = 1. Find infinitely many integer solutions to 3n + 5m = 1. In doing so, try to write down a general form for the solution, using the parameter k. Your answer should be of the form n = "an formula with k in it" and m = "a different formula with k in it".