2023 AMC 12A Problem 22

Problem 22 of 25HarderAlgebra Number Theory

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)?

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Solution

Setting

n=1

gives

f(1)=1.

For a prime

p,

n=p

gives

f(p)+p\cdot f(1)=1,

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,

\begin{gathered} f(2023)=f(7)\cdot f(17^2)\\ {}=(1-7)(1-17)\\ {}=(-6)(-16)\\ {}=96. \end{gathered}

Thus, the correct answer is B.