Let f be the unique function defined on the positive integers such that \sum_d\mid n d× f((n)/(d))=1 for all positive integers n, where the sum is taken over all positive divisors of n. What is f(2023)?

2023 AMC 12A Problem 22 — Solution

2023 AMC 12A Problem 22

Below is the professionally curated solution for Problem 22 of the 2023 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12A solutions, or check the answer key.

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Concepts:prime factorization recursion

Difficulty rating: 2270

22.

Let $f$ be the unique function defined on the positive integers such that $\sum_{d\mid n} d\cdot f\left(\frac{n}{d}\right)=1$ for all positive integers $n,$ where the sum is taken over all positive divisors of $n.$ What is $f(2023)?$

$-1536$

$96$

$108$

$116$

$144$

Solution:

Setting $n=1$ gives $f(1)=1.$ For a prime $p,$ $n=p$ gives $f(p)+p\cdot f(1)=1,$ so $f(p)=1-p.$ For $n=p^2,$ $f(p^2)+p\,f(p)+p^2 f(1)=1$ gives $f(p^2)=1-p.$

Since the defining relation is a Dirichlet convolution of multiplicative functions, $f$ is multiplicative. With $2023=7\cdot 17^2,$ $f(2023)=f(7)\cdot f(17^2)=(1-7)(1-17)=(-6)(-16)=96.$

Thus, the correct answer is B.

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