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)

Question

Let f be a function such that f(mn) = f(m) &times; 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

Accepted Answer

12 — Easy The rule $f(mn)=f(m)f(n)$ means $f$ multiplies across factorisations, so $f$ is determined by its values on primes $2$ and $3$. Use $f(24)=54$ to pin those down, then assemble $f(18)$. 1 Express $f(24)$ via primes. $24=2^{3}	imes 3$, so the rule gives $$\begin{aligned} &f(24)=f(2)^{3}\,f(3)=54 \end{aligned}$$ 2 Match against $54=2	imes 3^{3}$. We need positive integers $f(2),f(3)$ with $f(2)^{3}f(3)=54$. Trying $f(2)=3,\ f(3)=2$ gives $27	imes 2=54$. The condition $f(1) $$\begin{aligned}

CAT 2019 Slot 2 — QA Question 10

Answer & solution

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