Do a two-part proof and use either direct or proof by contrapositive.
Let b, n be positive integers where n is prime. Then b|(4b+n) if and only if b=1 or b=n.
I started with the if direction: Assume b=1 or b=n. Prove that b|(4b+n) Case 1: Assume b=1 1|4+n We know that n is a positive prime integer, therefore b|(4b+n) Case 2: Assume b=n Then b|(4b+n)=n|5n therefore b|(4b+n)
I'm getting stuck on the Only If direction. Assume b|(4b+n) Prove that b=1 or n. Then bk=4b+n and k=4+$\frac{n}{b}$ Can I now state that we know that since n is a positive prime number, it is only divisible by b if b is n or 1? Or is this against the rules.
