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Algebra Difficulty 3.7 AMC 10/12 Find the answer Canada

A function, $f$ , has $f(2)=5$ and $f(3)=7$ . In addition, $f$ has the property that $f(m)+f(n)=f(mn)$ for all positive integers $m$ and $n$ . (For example, $f(9) = f(3)+f(3) = 14$ .) The value of $f(12)$ is

$17$
$35$
$28$
$12$
$25$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $f(2) = 5$ and $f(mn)=f(m)+f(n)$ , then $f(4) = f(2 \cdot 2) = f(2) + f(2) = 10$ .

Since $f(3) = 7$ , then $f(12) = f(4 \cdot 3) = f(4) + f(3) = 10 + 7 = 17$ .

While this answers the question, is there actually a function that satisfies the requirements? The answer is yes.

One function that satisfies the requirements of the problem is the function $f$ defined by $f(1) = 0$ and $f(2^p 3^q r) = 5p+7q$ for all non-negative integers $p$ and $q$ and all positive integers $r$ that are not divisible by 2 or by 3. Can you see why this function satisfies the requirements?

Want a route through all this instead of an archive? The track puts 2,444 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.