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Sunday, December 27, 2015

2015/113) There is a number n between to successive squares.this is k larger than the smaller number and l smaller than the larger number . Prove that n-kl is a perfect square.

Let the number n be between a^2 and (a+1)^2
As per given condition
n- a^2 = k\cdots(1)
(a+1)^2 –n = l\cdots(2)
Adding (1) and (2)
k + l = 2a + 1
or l = 2a + 1 - k
now
n- kl = (a^2+k) – k(2a+1-k) = a^2 + k – 2ak –k + k^2 = a^2 – 2ak + k^2 = (a-k)^2
Hence proved

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