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Difference between revisions of "2021 AMC 12A Problems/Problem 25"

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==Problem==
 
==Problem==
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These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
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Let <math>d(n)</math> denote the number of positive integers that divide <math>n</math>, including <math>1</math> and <math>n</math>. For example, <math>d(1)=1,d(2)=2,</math> and <math>d(12)=6</math>. (This function is known as the divisor function.) Let<cmath>f(n)=\frac{d(n)}{\sqrt [3]n}.</cmath>There is a unique positive integer <math>N</math> such that <math>f(N)>f(n)</math> for all positive integers <math>n\ne N</math>. What is the sum of the digits of <math>N?</math>
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<math>\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9</math>
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==Solution==
 
==Solution==
 
Start off with the number 1. Multiply 1 by 2. When you multiply 1 by 2, you double the number of divisors and you divide it by <math>\sqrt[3]{2}</math>. Multiply 2 by 2. When you multiply 2 by 2, you triple and then halve the number of divisors. You must not forget to divide it by <math>\sqrt[3]{2}</math> again to get that function. Keep up multiplying by 2 until it starts to decrease. Keep on multiplying by 3 until it starts to decrease. Keep on multiplying by 5 until it starts to decrease. Keep on multiplying by 7 until it starts to decrease. You cannot multiply by 11 because first multiplication automatically causes a decrease. <math>1 \rightarrow 2 \rightarrow 2^2 \rightarrow 2^3 \rightarrow 2^33 \rightarrow 2^32^2 \rightarrow 2^33^25 \rightarrow2^33^257</math> This is like the computer science algorithms.  
 
Start off with the number 1. Multiply 1 by 2. When you multiply 1 by 2, you double the number of divisors and you divide it by <math>\sqrt[3]{2}</math>. Multiply 2 by 2. When you multiply 2 by 2, you triple and then halve the number of divisors. You must not forget to divide it by <math>\sqrt[3]{2}</math> again to get that function. Keep up multiplying by 2 until it starts to decrease. Keep on multiplying by 3 until it starts to decrease. Keep on multiplying by 5 until it starts to decrease. Keep on multiplying by 7 until it starts to decrease. You cannot multiply by 11 because first multiplication automatically causes a decrease. <math>1 \rightarrow 2 \rightarrow 2^2 \rightarrow 2^3 \rightarrow 2^33 \rightarrow 2^32^2 \rightarrow 2^33^25 \rightarrow2^33^257</math> This is like the computer science algorithms.  

Revision as of 16:05, 11 February 2021

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 1. Multiply 1 by 2. When you multiply 1 by 2, you double the number of divisors and you divide it by $\sqrt[3]{2}$. Multiply 2 by 2. When you multiply 2 by 2, you triple and then halve the number of divisors. You must not forget to divide it by $\sqrt[3]{2}$ again to get that function. Keep up multiplying by 2 until it starts to decrease. Keep on multiplying by 3 until it starts to decrease. Keep on multiplying by 5 until it starts to decrease. Keep on multiplying by 7 until it starts to decrease. You cannot multiply by 11 because first multiplication automatically causes a decrease. $1 \rightarrow 2 \rightarrow 2^2 \rightarrow 2^3 \rightarrow 2^33 \rightarrow 2^32^2 \rightarrow 2^33^25 \rightarrow2^33^257$ This is like the computer science algorithms. ~Lopkiloinm

Note

See problem 1.

See also

2021 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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