2024 AMC 12A Problem 3

Attempt Problem 3 of the 2024 AMC 12A 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 2024 AMC 12A solutions, or check the answer key.

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The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

$20$

$21$

$22$

$23$

$24$

Answer: B

Concepts:optimization bounding to limit cases

Difficulty rating: 1100

Solution:

Each number is at most $99,$ so $k$ numbers sum to at most $99k.$ Since $99\cdot20=1980\lt2024,$ at least $21$ numbers are required. With $21,$ we can use twenty $99$ s and one $44$ : $20\cdot99+44=1980+44$ $=2024.$ Thus, the correct answer is B.

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