Suppose $\displaystyle a,b$ are two integers with $\displaystyle \gcd(m,n)=1$. Prove that there exists integers $\displaystyle m,n$ such that $\displaystyle a^{m}+b^{n} \equiv 1 \mod{ab} $
$\displaystyle a^{\varphi(b)} + b^{\varphi(a)}$ is conguent to 1 mod a and also mod b, and hence mod ab ($\displaystyle \varphi$ is the Euler phi function).
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