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12 Sep 2011 01:22:53 IST

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Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Algebra

Prove that none of the integers 11, 111, 1111, 11111 .........is a square of an integer.

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hemang

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Joined: 27 Dec 2010

Posts: 1508

12 Sep 2011 11:52:17 IST

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Proof by contradiction

Suppose there exists an integer x such that x^2 = 111.....11.
x must be an odd integer because an even x can't have the ending digit 1.
Let's write x in the form x=2k+1 where k is an integer number.

Thus,
x^2 = 111.....11
=> (2k+1)^2 = 111...11
=> 4k^2 + 4k + 1 = 111...11
=> 4k^2 + 4k = 111...10
=> 4k^2 + 4k = 111...1 * 10
=> 4k(k+1) = 111...1 * 10
=> 2 k(k+1) = 555...5

The left hand side is an even number and the right hand side an odd number
which is a contradiction. Therefore there exists no integer x such that
x^2 = 111....11.

Hari Shankar

Forum Expert

Joined: 28 Feb 2007

Posts: 2185

12 Sep 2011 17:14:38 IST

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all the given numbers are of the form 4k+3 and no square is of this form

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