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Problem 667

AIME late

Number theory Difficulty 5.2 Find the answer

For each positive integer $n$ , an non-negative integer $f(n)$ is associated in such a way that the following three rules are satisfied:
i) $f(a b)=f(a)+f(b)$ .

ii) $f(n)=0$ if $n$ is a prime greater than 10.

iii) $f(1)<f(243)<f(2)<11$ .

Knowing that $f(2106)<11$ , determine the value of $f(96)$ .

Your answer

Official solution

Solution

By property $i$ ), we have $f(243)=f\left(3^{5}\right)=5 f(3)$ . Given that

$0 \leq f(1)<5 f(3)<f(2)<11$

and that $5 f(3)$ is a multiple of 5, we have $5 f(3)=5$ , that is, $f(3)=1$ . Note that $2106=2 \cdot 3^{4} \cdot 13$ . Thus, by property $i$ ),

$f(2106)=f(2)+4 f(3)+f(13)=f(2)+4$

From $f(2106)<11$ , it follows that $f(2)<7$ . Using $i$ i), we have $5<f(2)<7$ and thus $f(2)=6$ . Therefore,

$f(96)=f\left(3 \cdot 2^{5}\right)=f(3)+5 f(2)=31$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.