General Behavior of Euler Totient Function

Question

If we have two integer M and N such that

$$GCD(M,N) = k$$

Then what is

$$\phi(MN)$$

There is a famous identity which states:

$$GCD(M,N)= 1 \rightarrow \phi (MN) = \phi(M)\phi(N)$$

And now I am curious about generalizing this result to other numbers. Is there any literature on this problem?

Answer 1

Let $\gcd(m,n)=k$. Then the following identity is valid:

$$\varphi(m\cdot n)=\varphi(m)\cdot\varphi(n)\cdot\frac{k}{\varphi(k)}$$
If you want to learn more about Euler's totient function you could have a look here.