A contradiction about the Euler's totient function?

Question

$\phi(77) = \phi(7) \cdot \phi(11) = (7 - 1)(11 - 1) = 60$

but

$\phi(25) = \phi(5) \cdot \phi(5) = (5 - 1)(5 - 1) = 16 \neq 20$

What am I doing wrong?

Answer 1

Note that $$\phi(mn)=\phi(m)\phi(n)$$ holds only when $m$ and $n$ are relatively prime.

In your case $5$ and $5$ are not relatively prime so it does not apply.

You are dealing with $$\phi(p^2)=p(p-1)$$ formula.

Answer 2

You're assuming the totient function is fully multiplicative, it's not, mwahahaha! As you know, the numbers $1$ to $4$ are coprime to $5$. So are the numbers from $6$ to $9$, $11$ to $14$, $16$ to $19$ and $21$ to $24$, so clearly $\phi(5^2)$ is a multiple of $5$. In fact, $\phi(5^n)$ for $n > 1$ is always a multiple of $5$, which is not the case for $\phi(5m)$ when $\gcd(5, m) = 1$.

This can of course be generalized to other powers of odd primes, and multiples of thereof.