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I mean, suppose $$ab \equiv 1 \mod{m}$$ $$ac \equiv 1 \mod{n}$$ I wonder if there is any relation between $b$ and $c$? Could we compute one from another?

Thanks in advance!

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  • $\begingroup$ There is no relation if $m$ and $n$ are relatively prime. Of course if $d = (m,n) > 1$ then one necessary relation is that $b \equiv c \bmod d$. (That's also true if $d = 1$, but then it would not say anything serious.) $\endgroup$
    – KCd
    Apr 14, 2013 at 1:35
  • $\begingroup$ Hint: Try small numbers. $\endgroup$
    – Inceptio
    Apr 14, 2013 at 1:35
  • $\begingroup$ Look for the Chinese Remainder Theorem, it covers just this case. It says essentially that if $\gcd(m, n) = 1$, $a$ can be anything in $1 \le a \le m n$. $\endgroup$
    – vonbrand
    Apr 14, 2013 at 1:40

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