Question

Let $A=\{1,2,3,5,8,9\}$ Then the number of possible functions $f: A \rightarrow A$ 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______

Answer 1

Correct Answer: 432

$f(1)=1;f(9)=f(3)\times f(3)$
i.e., $f(3)=1$ or 3
Total function $=1\times 6\times 2\times 6\times 6\times 1=432$
So, the correct answer is 432.

Answer 2

Given :
f(1) = f(1 x 1) = f(1) x f(1) ⇒ f(1) = 1
f(9) = f(3x 3) = f(3) x f(3) ⇒f(3) = 1, f(9) = 1 or f(3) = 3, f(9) = 9
If f(1) = 1, f(3) = 1, f(9) = 1
Then f(2), f(5), f(8) each mapped in 6 ways
∴ Number of ways = 6³ = 216
∴ Number of ways = 6³ = 216
Total Ways =  216 + 216 = 432
So, the correct answer is 432.