Quandaries and Queries


Who is asking: Student
Level: Middle

Amy selected a positive integer N and wrote numbers 2xN+1 and 3xN+9 on a whiteboard. Boris looked at the two numbers on the whiteboard and wrote in his notebook a positive integer that divides each of them. Find all possible numbers that Boris could have written in his notebook.

Note: x means multiplied by.




What a neat problem! But why is it that they have to mess it up by bringing in Amy and Boris. Why not just say "find all possible common divisors of 2n+1 and 3n+9"?

Clearly 1 divides them both, but that's not very exciting. The key is to first look for primes that divide 2n+1 and 3n+9.

If the prime p were a common divisor, then 2n+1 = kp, which makes 3n+9 equal to 3(kp + 5)/2. What primes divide 3(kp + 5)?

3 divides 3(kp + 5). What other prime might divide 3(kp + 5)?

Chris and Penny


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