JEE Mathematics 2023 Question with Solution

Given: $A = \{1,2,3,5,8,9\}$ and $f : A \to A$ satisfies

for every $m,n \in A$ with $mn \in A$.

Find: The number of such functions.

From the condition with $1 \cdot 1 = 1$,

So,

Also, since $3 \cdot 3 = 9 \in A$,

Hence $f(3)^2 = f(9)$, and from the given solution this gives two possibilities:

corresponding to $f(9)=1$ or $f(9)=9$.

Case 1: $f(3)=1$ and $f(9)=1$.

Then $f(2), f(5), f(8)$ can each be chosen in $6$ ways. So the number of functions in this case is

Case 2: $f(3)=3$ and $f(9)=9$.

Again, $f(2), f(5), f(8)$ can each be chosen in $6$ ways. So the number of functions in this case is

Therefore, total number of functions is

So, the required answer is $432$.

Given: The only multiplicative relations inside $A$ that matter here are $1\cdot1=1$ and $3\cdot3=9$.

Find: Count all functions satisfying the condition.

First use

which forces

Next use

Since values of the function lie in $A = \{1,2,3,5,8,9\}$, the solution identifies only two admissible outcomes:

Once these are fixed, the elements $2,5,8$ are free to map anywhere in $A$, giving

functions for each admissible case.

Hence the total count is

Therefore, the answer is $432$.