2021 AMC 12B Fall Problem 12

Below is the professionally curated solution for Problem 12 of the 2021 AMC 12B Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Fall solutions, or check the answer key.

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Concepts:sum of factors prime factorization

Difficulty rating: 1760

12.

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, $f(14) = (1 + 2 + 7 + 14) \div 14 = \dfrac{12}{7}.$ What is $f(768) - f(384)?$

$\dfrac{1}{768}$

$\dfrac{1}{192}$

$1$

$\dfrac{4}{3}$

$\dfrac{8}{3}$

Solution:

Since $768 = 2^8 \cdot 3,$ its divisor sum is $(2^9 - 1)(1 + 3) = 511 \cdot 4 = 2044,$ so $f(768) = \dfrac{2044}{768} = \dfrac{511}{192}.$

Since $384 = 2^7 \cdot 3,$ its divisor sum is $(2^8 - 1)(1 + 3) = 255 \cdot 4 = 1020,$ so $f(384) = \dfrac{1020}{384} = \dfrac{510}{192}.$

The difference is $\dfrac{511 - 510}{192} = \dfrac{1}{192}.$

Thus, the correct answer is B.

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