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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 $b$.

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    Have you tried induction? If you can write $-k = 2a + 3b$ for some $a,b$, how would you write $-(k+1)$?2012-10-13
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    did not think to do that. Will try now, thanks. Also thanks for the edit.2012-10-13

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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 $n$.

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    that would probably have worked as well, but I did it with induction as the comment above suggested.2012-10-13
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    @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.2012-10-13
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    yeah, his is the answer I saw first, so I went that route. Thanks for your response though!2012-10-13
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    @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.)2012-10-13
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    I entirely agree! Thanks!2012-10-13