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Number Theory Level 3

Find least positive integer n such that

(n+20)+(n+21)+........(n+100) is a perfect square..


The answer is 4.

Naitik Sanghavi by naitik sanghavi , 20, India

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1 solution

Naitik Sanghavi
Jul 22, 2015

Here ,This is an Arithmetic Progression with first term =n+20 Last term=n+100 Number of terms=8

Therefore its sum is,

81/2×(n+20+n+100) =81(n+60)

So,for 81(n+60) to be a perfect square n+60=K^2

So least possible value of n=4..

3 Helpful 0 Interesting 0 Brilliant 0 Confused

small typo n=81 not 8.

Chirayu Bhardwaj - 5 years ago

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