2021 AMC 12A Problems/Problem 25

Revision as of 16:06, 11 February 2021 by Lopkiloinm (talk | contribs) (Solution)

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 $\frac{D(x)}{\sqrt[3]{x}}\frac{3}{\sqrt[3]{9}}$ If I am not mistaken, multiplying a number by 9 leads to an increase of a property of multiples of 9 is their digits add up to 9, so the only possibility is $\boxed{(D) 9}$ ~Lopkiloinm

Note

See problem 1.

See also

2021 AMC 12A (ProblemsAnswer KeyResources)
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Problem 24
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