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The question is the following:

"Determine the pair of numbers m,n such that gcd(1234,5678)=1234⋅m+5678⋅n for which n is the smallest positive integer".

I found that m=704 and n=-153. But n is negative here, so that can't be an answer. I assumed I could take 154 (since it's a multiple of 2) and add it to -153, which would give 1 and 1 is in ℤ2. However the pair (m,n)=(1,704) doesn't make much sense to me.

Why was my solution wrong and what is the right way to solve this problem?

Nare Avetisyan
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  • Welcome to Mathematics Stack Exchange. Hint: to continue your solution, add $\dfrac{1234}2 $ to $-153$, and subtract $\dfrac{5678}2$ from $704$. Do you see why that works? – J. W. Tanner Jan 21 '24 at 17:37
  • $2 = 1234 m+ 5678 m \iff 1 = \color{#c00}{617}m + 2839 n \iff n \equiv 2839^{-1} \pmod{\color{#c00}{617}}.,$ You computed $,n = -153,,$ whose least positive residue $\bmod \color{#c00}{617},$ is $,617-153 = 464\ \ $ – Bill Dubuque Jan 21 '24 at 20:29

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