perplexus.info :: Just Math : Oblong Difference Observation

Oblong Difference Observation (Posted on 2015-10-15)

Find all possible pairs (M, N) of oblong numbers that satisfy:

M - N = 2016

Prove that there are no others.

*** As an extra challenge solve this puzzle without using a computer program aided method.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

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solution

| Comment 1 of 5

I've never heard of oblong numbers before but using the linked definition M=m(m+1) and N=n(n+1).

Substitute into M-N=2016, multiply by 4, add and subtract 1 to get

(2m+1)^2 - (2n+1)^2 = (2m+2n+2)(2m-2n) = 8064 = (2^7)(3^2)(7) with solutions (m,n)=(47,15),(58,37),(116,107),(1008,1007).

Then solutions (M,N)=(2256,240),(3422,1406),(13572,11556), (1017072,1015056).

Posted by xdog on 2015-10-15 12:48:34

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