The six-digit number 2 0 2 1 0 A is prime for only one digit A. What is A?

2021 AMC 12A Fall Problem 4

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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

The number is

202100 + A;

even

A

makes it even, and

A = 5

makes it a multiple of

5

For

A = 1

and

A = 7

the digit sum is a multiple of

3,

and

202103 = 11 \cdot 18373

The number is

202100 + A.

Any even

A

makes it even, and

A = 5

makes it divisible by

5,

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.