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Math Help - (Perfect) Squares

  1. #1
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    (Perfect) Squares

    1.) Prove that a square of some integer has to leave a remainder of either 0 or 1 when it is divided by 4.


    2.) Prove that there is no number in the following sequence: 11, 111, 1111, 11111, ... that's a perfect square.
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  2. #2
    Junior Member AlvinCY's Avatar
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    1. Let the number you're after be p. There are only 4 possible senarios in modulo 4.

    p = 0 (mod 4) => p^2 = 0 (mod 4)
    p = 1 (mod 4) => p^2 = 1 (mod 4)
    p = 2 (mod 4) => p^2 = 4 = 0 (mod 4)
    p = 3 (mod 4) => p^2 = 9 = 1 (mod 4)

    Therefore any square number would have the property that p = 0 or 1 (mod 4)


    2. As you can see, you can write each of the 1111.....11 (n digits) numbers as:

    (n-2) digits ---> 111...111 x 100 + 11

    In mod 4, this is the same as 111...111 x (4 x 25) + 11 = 11 = 3 (mod 4)

    Since 111...111 is not equal to 0 or 1 mod 4, it cannot be a perfect square (from 1)
    Last edited by AlvinCY; February 20th 2007 at 12:24 AM.
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  3. #3
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    Quote Originally Posted by Ideasman View Post

    2.) Prove that there is no number in the following sequence: 11, 111, 1111, 11111, ... that's a perfect square.
    This one is simple.

    Because,
    111..1111 = 111....108 +3

    But 111....108 is divisible by 4 for the last two digits are.
    Thus, it is expressible as 4k.

    Thus,
    111....1111=4k+3 for some k in Z.

    But by the previous discussion squares are of the form 4k or 4k+1. Impossible.
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