2021 AMC 10A Spring Problem 11

Attempt Problem 11 of the 2021 AMC 10A Spring below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10A Spring solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

11.

For which of the following integers $b$ is the base- $b$ number $2021_b - 221_b$ not divisible by $3?$

$3$

$4$

$6$

$7$

$8$

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We can express this expression in base $10$ using the definition of bases: $2b^3 + 2b + 1 - 2b^2 - 2b - 1$ $= 2b^3 - 2b^2 = 2b^2(b - 1).$

For this to be divisible by $3,$ either $b$ or $b - 1$ must be divisible by $3.$

The only answer choice that satisfies neither of these conditions is $8.$

Thus, E is the correct answer.