Let a be an multiplicative arithmetic function. Show: either $a(1) = 1$ or $a(1) = 0$. Show that if $a(1) = 0$ then $a(n) = 0$ for all $n$

Question

I am asked to prove

Let a be an arithmetic function which is multiplicative. Show that either $a(1) = 1$ or $a(1) = 0$. Show that if $a(1) = 0$ then $a(n) = 0$ for all n.

From definition of arithmetic function, the output of the function is either $0$ or $1$, are there any way that i can actually proving that?

And for the second bit of the question, i tried couple of funtion such as the set of prime number; the set of squares number; the set of square-free numbers; but anyone can give me some hints of how to prove if $a(1) = 0$ then $a(n) = 0$ for all n.

Thanks

Answer 1

So $f:\mathbb{N}\to\mathbb{R}$ is such that $f(a\cdot b)=f(a)\cdot f(b)$ if $a$ and $b$ are relatively prime.

Let $x = a(1)$, then $$x=a(1)=a(1\cdot 1)=a(1)\cdot a(1)=x^2\implies x\in\{0,1\}$$

If $a(1)=0$ then $$a(n) = a(n)\cdot a(1) =0$$