2024 AMC 10A Problem 18

Below is the professionally curated solution for Problem 18 of the 2024 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 10A solutions, or check the answer key.

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Concepts:number base modular arithmetic counting integers in a range

Difficulty rating: 1840

18.

There are exactly $K$ positive integers $b$ with $5 \le b \le 2024$ such that the base- $b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K?$

$16$

$17$

$18$

$20$

$21$

Solution:

In base $b,$ $2024_b = 2b^3 + 2b + 4 = 2(b^3 + b + 2),$ so $16 \mid 2024_b$ exactly when $8 \mid b^3 + b + 2.$ Test the residues modulo $8:$ this holds precisely for $b \equiv 3, 6, 7 \pmod 8.$ Counting the $b$ with $5 \le b \le 2024$ in those three classes gives $K = 758,$ whose digit sum is $7 + 5 + 8 = 20.$ Therefore, the answer is D.

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