Question

Let $S=\{1,2,3,4,5,6,7\}$. Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to____.

Answer 1

$F(m n)=f(m) \cdot f(n)$
Put $m=1 f(n)=f(1) \cdot f(n) \Rightarrow f(1)=1$
Put $m=n=2$
$f(4)=f(2) \cdot f(2)\begin{cases}f(2)=1 \Rightarrow f(4)=1 \\ \text { or } \\ f(2)=2 \Rightarrow f(4)=4\end{cases}$
Put $m=2, n=3$
$f(6)=f(2) \cdot f(3)\begin{cases}\text { when } f(2)=1 \\ f(3)=1 \text { to } 7 \\ f(2)=2 f(3)=1 \text { or } 2 \text { or } 3\end{cases}$
$f(5), f(7)$ can take any value
Total $=(1 \times 1 \times 7 \times 1 \times 7 \times 1 \times 7)+(1 \times 1 \times 3 \times 1 \times 7 \times 1 \times 7)$
$=490$