Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions f : A A such that f ( m · n ) = f ( m ) · f ( n ) for every m , n A with m · n A is equal to __________________ . [2023]

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Solution

Q.
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 __________________ . [2023]

Ans.
(432)

Let $A = {1, 2, 3, 5, 8, 9}$

$f (m \cdot n) = f (m) \cdot f (n) for every m, n \in A and m \cdot n \in A$

Clearly, $f (1) = 1$ , i.e. $f (3) = 1$ or $3$

Total functions = $1 \times 6 \times 2 \times 6 \times 6 \times 1 = 432$

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