2021 Fall AMC 12B Problem 12

Problem 12 of 25IntermediateNumber Theory

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)?

Show solution

Solution

Since

768 = 2^8 \cdot 3,

its divisor sum is

(2^9 - 1)(1 + 3)

= 511 \cdot 4 = 2044,

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,

f(384) = \dfrac{1020}{384} = \dfrac{510}{192}.

The difference is

\dfrac{511 - 510}{192} = \dfrac{1}{192}.

Thus, the correct answer is B.