Multiplicative function value

Question

Let $f$ be a function such that $f(mn) = f(m)f(n)$ for every positive integers $m$ and $n$ . If $f(1), f(2)$ and $f(3)$ are positive integers, $f(1) < f(2)$ , and $f(24) = 54$ , then $f(18)$ equals

[TITA]

Accepted Answer

12. Given $f(1) = f(1 \cdot 1) = f(1)^2$. Since $f(1)$ is a positive integer, $f(1) = 1$. We are given $f(24) = 54$. Using the multiplicative property: $$f(24) = f(2^3 	imes 3) = [f(2)]^3 	imes f(3) = 54$$ Factorize $54 = 3^3 	imes 2$. Since $f(2) > f(1) = 1$ and $f(2), f(3)$ are integers, we must have $f(2) = 3$ and $f(3) = 2$. Now, calculate $f(18)$: $$f(18) = f(2 	imes 3^2) = f(2) 	imes [f(3)]^2 = 3 	imes 2^2 = 12$$

Answer

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Multiplicative function value

Explanation

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