# Any negative integer can be expressed as $2a+3b$

I am confused about a homework problem I have, and don't really know where to begin. The statement is that every negative integer can be written as $2a+3b$ where $a$ and $b$ are either positive or negative integers. I need to prove this. Any idea of where I can start. I am not necessarily looking for a solution, but a place to begin.

Show that every negative integer can be written in the form $2a + 3b$ for some (not necessarily positive) integers $a$ and $b$.

• Have you tried induction? If you can write $-k = 2a + 3b$ for some $a,b$, how would you write $-(k+1)$? – Austin Mohr Oct 13 '12 at 19:33
• did not think to do that. Will try now, thanks. Also thanks for the edit. – MZimmerman6 Oct 13 '12 at 19:38

## 1 Answer

HINT: First find integers $a_0$ and $b_0$ such that $$2a_0+3b_0=-1\;.\tag{1}$$ Then let $n$ be any positive integer, and see what happens when you multiply equation $(1)$ by $n$.

• that would probably have worked as well, but I did it with induction as the comment above suggested. – MZimmerman6 Oct 13 '12 at 20:55
• @MZimmerman6: Easy as it is in this case, induction actually takes a little more work. Since $2(1)+3(-1)=-1$, it’s immediate that $-n=2n+3(-n)$ for any positive integer $n$. This also gives you a specific representation of each negative integer. – Brian M. Scott Oct 13 '12 at 20:58
• yeah, his is the answer I saw first, so I went that route. Thanks for your response though! – MZimmerman6 Oct 13 '12 at 21:02
• @MZimmerman6: You’re welcome. (I wasn’t complaining, by the way; induction’s a perfectly reasonable approach. I just figured that since you’d now solved the problem, I might as well finish off the other solution, since it never hurts to see more than one approach.) – Brian M. Scott Oct 13 '12 at 21:05
• I entirely agree! Thanks! – MZimmerman6 Oct 13 '12 at 22:11