4.3. Find all functions $f: N \rightarrow N \cup\{0\}$ that satisfy the following conditions:

1) $f(m n)=f(m)+f(n)$ for all $m, n \in N$;
2) $f(10 m+3)=0$ for all $m \in N$;
3) $f(10)=0$.
4.3. $f(10)=0=f(5)+f(2) \rightarrow f(2)=f(5)=0$. Any natural number $n$ can be represented in the form $n=2^{k} 5^{l} b$, where $k, l \geq 0, (b, 10)=1$. Then, $b=10 m \pm 1$ or $10 m \pm 3$, so $b^{2}=10 m_{1} \pm 1 \Rightarrow b^{4}=10 m_{2}+1$.

Therefore, the number $3 b^{4}$ ends in the digit 3, so $f\left(3 b^{4}\right)=0, \Rightarrow f(3)=f(b)=0 \rightarrow f(n)=k f(2)+l f(5)+f(b)=0$.