Proof that any $\,a \ge 20\,$ can be written as $\,a = 5b + 6c\,$ where $\,a,\, b\,$ and $\,c \,$ are Natural Numbers

Question

I'm currently working on a problem for an assignment, so forgive me if my question is a little clumsy or vague. I'm trying to get myself headed in the right direction and offer a meaningful question here without just being handed the answer.

I'm trying to prove that $\,\forall\,a\in\mathbb{N}: \left[a \geq 20 \wedge\exists b,c \in\mathbb{N}: a = 5b + 6c\right] \rightarrow\exists\; b,\,c \in\mathbb{N}: a + 1 = 5b + 6c$

I really haven't a clue where to begin. I would appreciate any hint or suggestion as to an approach.

Answer 1

Hint: If you can prove that $20,21,22,23,24$ can each be written in such a way, then you can add one extra five to each of those and thereby represent $25, 26,27,28,29$.