2021 AMC 12B Spring Problem 10

Below is the professionally curated solution for Problem 10 of the 2021 AMC 12B Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

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Concepts:Simon’s Favorite Factoring Trick arithmetic sequence

Difficulty rating: 1530

10.

Two distinct numbers are selected from the set $\{1,2,3,4,\ldots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?

$5$

$7$

$8$

$9$

$10$

Solution:

The sum $1+2+\cdots+37=703.$ If the chosen numbers are $a$ and $b,$ then $703-a-b=ab.$

So $ab+a+b=703,$ and adding $1$ gives $(a+1)(b+1)=704=2^6\cdot 11.$

We need factors $a+1,b+1$ between $2$ and $38.$ The pair $22\cdot 32=704$ works, giving $a=21,$ $b=31.$

Their difference is $31-21=10.$

Thus, the correct answer is E.

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