2024 AMC 12A Problem 12

Below is the professionally curated solution for Problem 12 of the 2024 AMC 12A, 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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Concepts:geometric sequence factor optimization

Difficulty rating: 1630

12.

The first three terms of a geometric sequence are the integers $a,\ 720,$ and $b,$ where $a\lt720\lt b.$ What is the sum of the digits of the least possible value of $b?$

$9$

$12$

$16$

$18$

$21$

Solution:

Since the terms are geometric, $720^2=ab,$ so $ab=518400=2^8\cdot3^4\cdot5^2.$ Because $b=518400/a,$ minimizing $b$ means maximizing the divisor $a$ subject to $a\lt720.$ The largest such divisor is $a=675=3^3\cdot5^2,$ giving $b=518400/675=768.$ Its digit sum is $7+6+8=21.$ Thus, the correct answer is E.

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