2021 AMC 12B Fall Problem 12

Attempt Problem 12 of the 2021 AMC 12B Fall below, then check your answer against the professionally curated solution 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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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,

\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

Small hint:

Factor

768 = 2^8 \cdot 3

and

384 = 2^7 \cdot 3,

then use

\sigma

of a prime power

Big hint:

Put both over a common denominator of

192

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.

Problem 12 in other years