JEE Main 2023 (Online) 30th January Evening ShiftQuestion 1
Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f:A\to A$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in A$ with $m\cdot n\in A$ is equal to
Functions
Answer$432$
Solution
Products inside $A$: $1\cdot n=n$ and $3\cdot3=9$. Then $f(1)=1$ and $f(9)=f(3)^2\in A$ forces $f(3)\in\{1,3\}$ (2 choices), while $f(2),f(5),f(8)$ are free (6 each). Count $=2\cdot6^3=432$.