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So this is the question I'm trying to find a solution to:

Prove or disprove whether you can order the years 1985, . . . , 1995 so that the resulting 44-digit number is prime.

I've tried some brute force methods to try to see a pattern, using divisibility rules to rule out possible numbers and trying to multiply prime numbers together to get one of the possible 44-digit numbers as a resulting composite number. It became clear pretty quickly how long it'd take to try using either method to solve this, so now I'm a little stuck. There has got to be a much more elegant method for solving this that I'm not seeing.

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    $\begingroup$ The number is always divisible by 11. In general in these type of trick problems the answer must be yes, and the way to proceed is to check mod 2, 3,5,11 $\endgroup$ – cdt Mar 19 at 2:36
  • $\begingroup$ To add to @alxchen's comment, those are the small primes with reasonable divisibility rules. 7 is only missing because it's a pain to check... $\endgroup$ – Michael Biro Mar 19 at 2:37
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The trick to use here is divisibility by 11. Since all of the years have 4 digits, you add all the thousands places, subtract the hundreds places, add the tens places, and subtract the ones places.

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