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, $f(9x) > f(x)$ . Multiplying a number by 9 leads to an increase and 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.

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2021 AMC 12A Problems/Problem 25

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