2021 AMC 12A Problems/Problem 25

Problem

Let $d(n)$ denote the number of positive integers that divide $n$ , including $1$ and $n$ . For example, $d(1)=1,d(2)=2,$ and $d(12)=6$ . (This function is known as the divisor function.) Let $f(n)=\frac{d(n)}{\sqrt [3]n}.$ There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$ . What is the sum of the digits of $N?$

$\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9$

Solution

Start off with the number x that does not have a factor of 3. Multiply x by 9. Multiplying x by 9 triples the number of divisors but also leads to a decrease in the function because of the denominator being multiplied by $\sqrt[3]{9}$ . The number is now $\frac{D(x)}{\sqrt[3]{x}}\frac{3}{\sqrt[3]{9}}$ . Multiplying a nonmultiple of 3 by 9 making a bigger f leads to this truth being known, $f(9x) > f(x)$ . A property of multiples of 9 is their digits add up to 9, so the only possibility is $\boxed{(E) 9}$ ~Lopkiloinm

Note

See problem 1.

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2021 AMC 12A Problems/Problem 25

Contents

Problem

Solution

Note

See also