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Euler's totient function $\varphi$ is multiplicative, and it plays important roles in number theory. QUESTION: Is it true that for each integer $m>6$ we have $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$...

Let $\varphi$ be Euler's totient function. If $p$ is a prime, then $\varphi(1+n)=n$ for $n=p-1$. Question. Is it true that for each integer $m>1$ there is a positive integer $n\le m^2-m$ such that ...