mathematical induction to prove $n=6x+13y$

Question

Prove using Mathematical Induction (without strong MI) that any number n >=60 can be expressed as $6 *x + 13 *y$ where x and y are non-negative.
My Thoughts:
Let $n= 6*x+13*y$ then $n+1 = 6*(x-2) + 13 * (y+1)$ so it can be expressed but then there is a possibility that $x-2<0$ which will be when $x=0$ or $x=1$. I am not sure how to proceed in this case

Answer 1

Your strategy makes sense to me. It basically works because you can write $1$ as $-2 \cdot 6 + 1 \cdot 13$. If $x \ge 2$, your proof works perfectly.

You could alternatively write $1$ as $11 \cdot 6 - 5 \cdot 13$. That means if $n = 6x + 13y$ then you could alternatively use $n+1 = 6(x+11) + 13(y-5)$.

Now, can you see how to use my new observation to finish the problem?