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# Need Quick Clarity On This Question

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So I have this math problem:

The solution of $8x+1 \equiv 5$ (mod $12$) is $x \equiv a$ (mod $m$) for some positive integers $m \geq 2$ and $a < m$. Find $a+m$.

I saw an answer that said $m$ is a multiple of $3$ and that $a=-1$. Does this mean that the answer can be $3-1=2$ or $6-1=5$ etc... or can I choose any one solution as my solution?

Jun 18, 2021

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I get a = -1 and m = 3, so a + m = 2.

Jun 18, 2021
#2
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The number a in such context is always taken to be a nonnegative integer less than m.  If you find a solution that is negative, do add an appropriate multiple of m to it to make it nonnegative and less than m.  E.g. if you find a solution of -2 for a, and your m is 5, then add 5 to a to get -2+5 = 3.  So your a is actually 3.   So a+m in this case is 3+5 = 8.

Jun 20, 2021