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(#M40005675) NTSE question No. System Keep an EYE

A natural No n is chosen strictly between two consecutive perfect squares.The smaller of these two squares is obtained by subtracting k from n and the larger one is obtained by adding L to n.Prove that n-kL is a perfect square.

Asked In NTSE Sanjana Patnaik (13 years ago)

Solved Pargat Pal Singh Read Solution (1)

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Pargat Pal Singh  (13 years ago)

Let Squares are p2 and (p + 1)2
given p^2 = n - k
(p + 1)2 = n + l
Now n- kl
= n - (n - p2) ((p + 1)^2 - n)
= n - n(p + 1)^2 + n^2 + p^2(p + 1)^2 - np^2
= n - np^2 -2np - n + n^2 - np^2 + p^2(p + 1)^2
= n2 - 2np^2 - 2np + p^2(p + 1)^2
= n2 - 2np(p + 1) + p^2(p + 1)^2
= [n - (p + 1)p]2
so n-kL is a perfect square.

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