Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 1I / Solution

If $m$ and $n$ are coprime, every divisor of $mn$ is uniquely $ab$ with $a\mid m$ and $b\mid n$. Hence, for a multiplicative arithmetic function $f$,

The Möbius function is $\mu (n)=0$ if a prime square divides $n$, and $\mu (n)=(-1{)}^r$ if $n$ is a product of $r$ distinct primes. The Euler totient function $\varphi (n)$ counts residues modulo $n$ coprime to $n$. From the prime factorizations,Both sides of the second identity are multiplicative, and at a prime power $p^a$,Therefore

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