Problem:
The six-digit number is prime for only one digit A . What is A ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
A number whose units digit is 5 is divisible by 5, so choice (C) does not give a prime. Choices (A) and (D) can be ruled out by the fact that a number is divisible by 3 if and only if the sum of its digits is
divisible by 3 ; here and are divisible by 3 . Choice (B) can be ruled out by the fact that a number is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 ; here is divisible by 11 . The five even choices for A result in a number divisible by 2 . Given that one choice produces a prime, A must equal and the number 202109 must be prime. (In fact, this can be verified with computer algebra software.)
The problems on this page are the property of the MAA's American Mathematics Competitions