JEE MAIN - Mathematics (2023 - 30th January Evening Shift - No. 14)
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
432
Explanation
$$f(1.n)=f(1).f(n)\Rightarrow f(1)=1$$.
$$f(3.3)=(f(3))^2$$
Hence, the possibilities for $$(t(3),(9))$$ are $$(1,1)$$ and $$(3,9)$$.
Other three i.e. $$f(2),f(5),f(8)$$
Can be chosen in 6$$^3$$ ways.
Hence, total number of functions
$$6^3\times2=432$$
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