How many maps $\phi:\Bbb{N}\to\Bbb{N}$ are there with $\phi(ab) = \phi(a)+\phi(b)$? I am stuck on the following problem when I was trying to solve an entrance exam paper:

How many maps $\phi \colon \Bbb N \cup \{0\} \to \Bbb N \cup \{0\}$ are there with the property that $\phi(ab)=\phi(a)+\phi(b)$ for all $a,b \in \Bbb N \cup \{0\}$ ?

The options are as follows :


*

*none


*finitely many


*countably many


*uncountably many

Option 1 is not possible as if we take $a=1,b=0$ then we get $\phi(1)=0$ which is possible. But, I am not sure about the other options and could not decide which one holds good.
Can someone explain?

Hint: Consider $\phi(0\cdot b)$.
