2021 AMC 12A Fall Problem 4

Attempt Problem 4 of the 2021 AMC 12A Fall 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 2021 AMC 12A Fall solutions, or check the answer key.

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

The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$

$1$

$3$

$5$

$7$

$9$

Answer: E

Concepts:prime divisibility

Difficulty rating: 1200

Solution:

The number is $202100 + A.$ Any even $A$ makes it even, and $A = 5$ makes it divisible by $5,$ so $A$ must be odd and not $5.$

For $A = 1$ the digit sum is $6$ (divisible by $3$ ); for $A = 7$ the digit sum is $12$ (divisible by $3$ ); and $202103 = 11 \cdot 18373.$ Only $202109$ survives all tests, and it is prime.

Thus, the correct answer is E.

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