Question

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 $A?$

$1$

$3$

$5$

$7$

$9$

Solution:

Note that $A$ cannot be even, as then the number would be divisible by $2.$

$A$ also cannot be $5,$ as that would make the number divisible by $5.$

If $A$ equaled $1$ or $7,$ then the sum of the digits of the number would be $6$ and $12$ respectively.

This would make the number divisible by $3,$ so that rules out $A$ equaling either of these numbers.

Finally, if $A$ equals $3,$ then the whole number becomes $202109.$ If we look at the difference of the sums of alternating digits, we get $2 + 2 - 1 - 3 = 0,$ which means the number is divisible by $11.$

This means that $A$ must be $9.$

Thus, E is the correct answer.

2021 AMC 10A Fall Problem 5

Solution:

Problem 5 in Other Years