2024 AMC 12A Problem 11

Below is the professionally curated solution for Problem 11 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:number base modular arithmetic counting integers in a range

Difficulty rating: 1630

11.

There are exactly $K$ positive integers $b$ with $5\le b\le2024$ 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:

Here $2024_b=2b^3+2b+4=2(b^3+b+2),$ so $16\mid2024_b$ exactly when $8\mid b^3+b+2.$ Checking residues $\bmod 8,$ $b^3+b+2\equiv0$ precisely for $b\equiv3,6,7\pmod8.$

Counting $b$ in $[5,2024]:$ residue $3$ gives $11,\ldots,2019$ ( $252$ values), residue $6$ gives $6,\ldots,2022$ ( $253$ values), and residue $7$ gives $7,\ldots,2023$ ( $253$ values). So $K=252+253+253=758,$ and its digit sum is $7+5+8=20.$

Thus, the correct answer is D.

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