Why f(1)=1 for every multiplicative function f?

Question

If $f$ is a multiplicative function with $f(1)\ne0$, then why is $f(1)$ necessarily equal to $1$?

Answer 1

Hint: Considers what happens to $f(1)f(1)$.

Answer 2

Since we know $f$ is multiplicative, then $f(ab)=f(a)f(b)$.

So $f(1\cdot a)= f(a) =f(1)f(a)$.

This is only true when $f$ is zero everywhere, or $f(1)=1$.