I wonder if you might get more responses if this were posted in the Number Theory subforum.

There is, however, a very easy answer. What's an easy test for divisibility by 3 based on a number's decimal digits?

EDIT: Sorry, I completely forgot what the question asked momentarily, and the above gives a reason for there being infinitely many composites. I'll have to revisit the question. (And I may not be able to find an answer.)

(By the way, does anyone know how to get strikethrough text formatting? I tried searching but couldn't find a way. So I made my wrong answer gray instead.)

Edit 2: If the right answer involves divisibility tests for small primes then this link ("Divisibility by prime numbers under 50" by Stu Savory) might be helpful (or might not).