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Perfect Square To Divisibility By 56 (Posted on 2010-04-01) Difficulty: 4 of 5
N is a positive integer such that each of 3*N + 1 and 4*N + 1 is a perfect square.

Is N always divisible by 56?

If so, prove it. Otherwise, give a counterexample.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: Modular arithmetic | Comment 10 of 14 |
(In reply to Modular arithmetic by Brian Smith)

Well done Brian, I think youíve moved us on somewhat, but Iím still not sure how this modulo 7 approach can be completed.

Incidentally, I notice that in your 4th paragraph you claim that N = 7x + 4 remains a possibility, but I think that should be N = 7x + 5, which then gives
4N + 1 = 28x + 21 rather than 4N + 1 = 28x + 17.
This means that both N = 8(7x + 2) and N = 8(7x + 5) still need to be ruled out to prove that N is a multiple of 56.

Iím beginning to think that the squared numbers themselves are the key to solving this problem. Iíll try to post something tomorrow.

  Posted by Harry on 2010-04-03 21:23:59

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