Find m and n where m + n = 72, and gcd (m , n) = 9
How to write an answer in this case, or smth wrong in the way i understand it? Because, (9, 63), (18, 54), (27, 45), etc, satisfy conditions.
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$\gcd(18,54)=18$, not $9$. But, in any case, sometimes equations have more than one solution. If you can, list all of them. – lulu Dec 22 '21 at 18:14
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2You need only to find $p$ and $q$ with $\text {gcd} (p,q) = 1$ and $p + q = 8.$ This gives you the possible ordered pairs $(p,q)$ which are $(1,7)$ and $(3,5).$ So the required ordered pairs $(m,n)$ are $(9p,9q) = (9,63)$ and $(27,45).$ So there are only two such pairs. – RKC Dec 22 '21 at 18:16
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my bad, missed the fact that gcd(18, 54) != 9. Thanks a lot – kertal Dec 22 '21 at 18:25
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As in the linked dupe, canceling the gcd reduces to the coprime case $,\bar m + \bar n = 8,\ \gcd(\bar m,\bar n) = 1,\ $ whose solution is straightforward, since $,\gcd(\bar m,\bar n)=\gcd(\bar m,8-\bar m) = \gcd (\bar m,8) = 1\iff m,$ is odd. – Bill Dubuque Dec 22 '21 at 18:36