2021 Fall AMC 12A Problem 4

Problem 4 of 25EasierNumber Theory

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?

Show solution

Solution

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.