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Modulus and Squares

  1. Dec 19, 2006 #1
    1. The problem statement, all variables and given/known data

    X mod m is the remainder when x is divided by m. This value is called a residue. Find all perfect squares from the set of residues mod 16.

    3. The attempt at a solution

    There was a suggestion that this would become clearer when the definition of perfect square was reviewed and found to be more than just a square rootable number.

    I can't find a better definition of a perfect square than a number that has a square root that is an integer. Can anyone point me to an exhaustive definition of a perfect square?

  2. jcsd
  3. Dec 20, 2006 #2

    matt grime

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    You have it. Let A be some system of arithmetic - the integers, or the integers mod m.

    s in A is a square if s=r^2 for some r in A.
  4. Dec 20, 2006 #3

    Thanks for the reply. One bit of clarification;

    a and b are integers and a^2 = b

    Is a the perfect square?

    So if I was asked to find the perfect square of b, then the answer would be a?

    Sorry for all the questions, but I'm struggling with the english syntax of this problem. I guess Shaw was right when he said the US and England were sperated by a common language. Heck, NY and CA are seperated by a common language, hahaha.

  5. Dec 20, 2006 #4

    matt grime

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    Erm, no there is no language separation here. Please reread what was written. I said s was a square (prefect, if you must) if s was equal/equivalent to r^2 for some r. So, if all else fails, to find the squares mod 16, all you need to do is take the 16 residues, 0,1,..,15 and square them all and see what you get. In fact that is probably the most sensible way to do the problem.
  6. Dec 21, 2006 #5
    Thanks Matt!

    I got it figured out. Along the way I also figured out there are many wrong definitions on the net for perfect squares, hahahaha.

  7. Dec 21, 2006 #6


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    There are exactly 16 members of the set of residues mod 16, 0 through 15.
    Square each of them and find the residue mod 16. For example, 22= 4 mod 16 so 4 is a perfect square mod 16. 32= 9 mod 16 so 9 is a perfect square mod 16. The "mod" part doesn't come in until you get numbers whose square is greater than 16: 52= 25= 16+ 9 = 9 mod 16 so that just gives you "9 is a perfect square mod 16" again. 62= 36= 2(16)+ 4 so "4 is a perfect square mod 16" again.
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