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 Square pairs (Posted on 2018-02-07)
Let's denote by p(n) the greatest square less than or equal to n,
n being a positive integer.

List all pairs (m,n), for which:(2m+1)*p(2n+1) = 400; m being less than n.

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 Answer Comment 3 of 3 |
400 is a square and p(2n+1) is a square. Since (2m+1)*p(2n+1)=400, then 2m+1 is a square that divides 400. Therefore, 2m+1=1 or 25. If 2m+1=1, then m=0. Then, p(2n+1)=400, so 400<=2n+1<441. Then, 200<=n<=219. If 2m+1=25, then m=12. Then, p(2n+1)=16, so 16<=2n+1<25. Then, 8<=n<=11. However, m<n, so this is impossible. Therefore, m=0 and 200<=n<=219.

Solutions:(0, 200), (0, 201), (0, 202), (0, 203), (0, 204), (0, 205), (0, 206), (0, 207), (0, 208), (0, 209), (0, 210), (0, 211), (0, 212), (0, 213), (0, 214), (0, 215), (0, 216), (0, 217), (0, 218), (0, 219)

 Posted by Math Man on 2018-02-07 21:27:56

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