Take all letters as integers. Let m=ax+b and n=cy+d For what values of x and y will m and n be equal, where a and c are relatively prime. Can we do this by getting the least value for m and n? For example is m= 5x+3 and n=3y+2 m and n are equal to 8, for x=1 and y=2. If we assume b and d are equal to zero, things seem to get simple and m=n=ac, x=c and y=a, but if we assume integers that for b and d when they are not zero, things get a bit tricky.
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1Please refer to the Euclidean Algorithm for first order Diophantine equations. There is a wealth of relevant content on this site. – Jam Oct 05 '23 at 15:42
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1If $d=b$, things are also easy. Then $ax=cy$, which has solutions $x=y=0$ or $x=c$, $y=a$ etc. – Dietrich Burde Oct 05 '23 at 16:18
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1$m = n\iff n \equiv b\pmod{!a},\ n \equiv d\pmod{!c}.,$ A formula for the general solution of this system of congruences is given by General Easy CRT in the linked dupe. – Bill Dubuque Oct 05 '23 at 17:18
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1Or $,m=n\iff ax-cy = d-b,$ which can be solved e.g. by the extended Euclidean algorithm, see here – Bill Dubuque Oct 05 '23 at 17:25