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Algebra

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 Joined: 12 Sep 2011 Post: 1
12 Sep 2011 01:22:53 IST
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Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

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

Blazing goIITian

Joined: 27 Dec 2010
Posts: 1508
12 Sep 2011 11:52:17 IST
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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.

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Joined: 28 Feb 2007
Posts: 2217
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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