Well, I know two or three proofs of this fact $$\gcd(m,n)=1\implies \varphi(mn)=\varphi(m)\varphi(n)$$ where $\varphi$ is the totient function.
My problem is this: I'd like to explain this to some gifted children. The children are gifted enough to understand some basic facts, like the reason why $\varphi(n)$ is even ($k$ and $n$ are coprime $\iff$ $n-k$ and $k$ are, so make pairs) and they have a very solid intuitive idea of the Fundamental Theorem of Arithmetic (they know, for example, that if $a$ and $b$ are coprime, and $ab$ is a square, then both $a$ and $b$ are squares).
But I have tried to taught them why the totient function is multiplicative, but I (and they) understand that $\varphi$ is multiplicative but not an intuitive reason.
How would you explain it?