2021 AMC 10B Fall Problem 7

Below is the professionally curated solution for Problem 7 of the 2021 AMC 10B Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10B Fall solutions, or check the answer key.

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Concepts:fraction divisibility systematic listing

Difficulty rating: 1660

Call a fraction $\frac{a}{b},$ not necessarily in the simplest form, ''special'' if $a$ and $b$ are positive integers whose sum is $15.$ How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

$\ 9$

$\ 10$

$\ 11$

$\ 12$

$\ 13$

Solution:

A special fraction with denominator $b$ equals $\frac{15-b}{b}=\frac{15}{b}-1$ , where $1\le b\le14$ . We need integer values of $\frac{15}{x}+\frac{15}{y}-2$ .

Checking the possible denominators by fractional part gives the distinct integer sums $1,2,3,4,6,7,8,13,16,18,28.$

There are $11$ such integers.

Thus, the answer is C .

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