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2021 Fall AMC 12B Problems/Problem 12

Revision as of 19:09, 26 January 2022 by MRENTHUSIASM (talk | contribs)

Problem

For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of n is divided by n. For example, \[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\] What is $f(768)-f(384)?$

$\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}$

Solution 1

The prime factorization of $768$ is $2^8\cdot3,$ and the prime factorization of $384$ is $2^7\cdot3.$ Note that \begin{alignat*}{8} f(768)&=\left(1+\frac{1}{2}+\cdots+\frac{1}{2^8}\right)+\left(\frac{1}{3}+\frac{1}{2\cdot3}+\cdots+\frac{1}{2^8\cdot3}\right)&&=\left(1+\frac{1}{2}+\cdots+\frac{1}{2^8}\right)\left(1+\frac{1}{3}\right)&&=\frac{511}{256}\cdot\frac{4}{3}&&=\frac{511}{192}, \\ f(384)&=\left(1+\frac{1}{2}+\cdots+\frac{1}{2^7}\right)+\left(\frac{1}{3}+\frac{1}{2\cdot3}+\cdots+\frac{1}{2^7\cdot3}\right)&&=\left(1+\frac{1}{2}+\cdots+\frac{1}{2^7}\right)\left(1+\frac{1}{3}\right)&&=\frac{255}{128}\cdot\frac{4}{3}&&=\frac{85}{32}. \end{alignat*} Therefore, the answer is \[f(768)-f(384)=\frac{511}{192}-\frac{85}{32}=\frac{511}{192}-\frac{510}{192}=\boxed{\textbf{(B)}\ \frac{1}{192}}.\] ~lopkiloinm ~MRENTHUSIASM

Solution 2

Let $\sigma(n)$ denotes the sum of the positive integer divisors of $n,$ so $f(n)=\frac{\sigma(n)}{n}.$

Suppose that $n=\prod_{i=1}^{k}p_i^{e_i}$ is the prime factorization of $n.$ Since $\sigma(n)$ is multiplicative, we have \[\sigma(n)=\sigma\left(\prod_{i=1}^{k}p_i^{e_i}\right)=\prod_{i=1}^{k}\sigma\left(p_i^{e_i}\right)=\prod_{i=1}^{k}\left(\sum_{j=0}^{e_i}p_i^j\right)=\prod_{i=1}^{k}\frac{p_i^{e_i+1}-1}{p_i-1}.\] The prime factorization of $768$ is $2^8\cdot3,$ and the prime factorization of $384$ is $2^7\cdot3.$ Note that \begin{alignat*}{8} f(768) &= \left(\frac{2^9-1}{2-1}\cdot\frac{3^2-1}{3-1}\right)\div768 &&= \frac{511}{192}, \\ f(384) &= \left(\frac{2^8-1}{2-1}\cdot\frac{3^2-1}{3-1}\right)\div384 &&= \frac{85}{32}. \end{alignat*} Therefore, the answer is \[f(768)-f(384)=\frac{511}{192}-\frac{85}{32}=\frac{511}{192}-\frac{510}{192}=\boxed{\textbf{(B)}\ \frac{1}{192}}.\] ~MRENTHUSIASM

Solution 3

We see that the prime factorization of $384$ is $2^7 \cdot 3.$ Each of its divisors is in the form of $2^x$ or $2^x \cdot 3$ for a nonnegative integer $x \le 7.$ We can use this fact to our advantage when calculating the sum of all of them. Notice that \[2^x + 2^x \cdot 3 = 2^x(1+3) = 2^x \cdot 4 = 2^x \cdot 2^2 = 2^{x+2}\] is the sum of the two forms of divisors for each $x$ from $0$ to $7,$ inclusive. So, the sum of all of the divisors of $384$ is just \[2^2 + 2^3 + 2^4 + \cdots + 2^9 = (2^0 + 2^1 + 2^2 + 2^3 + 2^4 + \cdots + 2^9) - (2^0 + 2^1) = (2^{10} - 1) - (2^0 + 2^1) = 1020.\] Therefore, we have $f(384) = \frac{1020}{384}.$ Similarly, since $768 = 2^8 \cdot 3,$ we have $f(768) = \frac{2044}{768}.$

Finally, the answer is \[f(768) - f(384) = \frac{2044}{768} - \frac{1020}{384} = \frac{2044}{768} - \frac{2040}{768} = \frac{4}{768} = \boxed{\textbf{(B)}\ \frac{1}{192}}.\]

~mahaler

See Also

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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

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