2019 AMC 10A Problem 2

Attempt Problem 2 of the 2019 AMC 10A 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 2019 AMC 10A solutions, or check the answer key.

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2.

What is the hundreds digit of (20!15!)?(20!-15!)?

00

11

22

44

55

Answer: A
Concepts:factorialtrailing zerosdivisibility
Difficulty rating: 770
Solution:

Both 20!20! and 15!15! contain at least three factors of 55 and three factors of 22, so both are divisible by 2353=1000.2^3\cdot5^3=1000.

Their difference is therefore also divisible by 1000.1000.

Being a multiple of 10001000 makes the last three digits 0,0, which shows that the hundreds digit is also 0.0.

Thus, A is the correct answer.

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