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.
I've never heard of oblong numbers before but using the linked definition M=m(m+1) and N=n(n+1).
Substitute into MN=2016, multiply by 4, add and subtract 1 to get
(2m+1)^2  (2n+1)^2 = (2m+2n+2)(2m2n) = 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 20151015 12:48:34 