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We need to find a minimum of $v$ in :

$$N = du + sv$$

where $N, d,s$ are given positive integer, $u$ and $v$ are arbitrary positive integers.

For example, $19 = 5\cdot 2 + 3\cdot 3$ where $d,u,s,v = 5, 2, 3, 3$ resp.

How can we find a minimum of such a $v$, and if it may not exist?

I know very basic modulo properties, but they do not give direct answer.

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