Question

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

$\dfrac{1}{768}$

$\dfrac{1}{192}$

$1$

$\dfrac{4}{3}$

$\dfrac{8}{3}$

Answer: B

Concepts:sum of factors prime factorization

Difficulty rating: 1760

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.

2021 AMC 12B Fall Problem 12

Answer: B

Difficulty rating: 1760

Solution:

Problem 12 in Other Years