2021 AMC 12B Spring Problem 10

Attempt Problem 10 of the 2021 AMC 12B Spring 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 Spring solutions, or check the answer key.

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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

The full set sums to

703;

if the two chosen numbers are

a

and

b,

then

703-a-b=ab

Rearrange to

(a+1)(b+1)=704

and factor

The sum

1+2+\cdots+37=703.

If the chosen numbers are

a

and

b,

then

703-a-b=ab.

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.

Problem 10 in other years