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Determine the remainders when dividing their squares by four

  1. Apr 29, 2007 #1
    Randomly select eight odd integers of less than 1000

    a) Determine the remainders when dividing their squares by four, and tabualte your results

    b) Make a conjecture about your findings

    c) test your conjecture with at least five larger integers

    d) Prove of justify the conjecture you make.


    n tn = n^2 / 4 - 1, Remainder
    49 600 1
    81 1640 1
    121 3660 1
    225 12656 1
    441 48620 1
    625 97656 1
    841 176820 1
    -----------------------------------------
    2001 1001000 1
    3673 3372732 1
    6925 11988906 1
    8123 16495782 1
    9999 249950000 1

    b) When the square of an odd integer, for n is equal to or greater than 1, is divided by four the result is an even integer with a remainder of 1.

    c)
    You know that tn = n^2 / 4 - 1, by hypothesis
    Factoring n, tn = (n x n)/4 -1

    I'm not quite sure where to take my proof from here. Could someone give me a nudge in the right direction?
     
  2. jcsd
  3. Apr 29, 2007 #2

    HallsofIvy

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    These are odd integers! Any odd integer can be written in the form
    2n+ 1. What is the square of that? what do you get when you divide by 4?
     
  4. Apr 29, 2007 #3
    EDIT:

    (2n+1)^2 / 4 = n^2 +n + 1/4 = n(n +1) + 1/4

    The remainder is 1/4
     
    Last edited: Apr 29, 2007
  5. Apr 29, 2007 #4
    (2n + 1)^2 is not 4n^2 + 1.

    And I think he was more concerned with the remainder you get when you divide by 4, not the expression in terms of fractions.
     
  6. Apr 29, 2007 #5
    Proof

    An odd integer is a multiple of 2, plus 1. Let 2n + 1 represent an odd integer.

    The square of an odd integer is 4n^2 + 4n +1 which when divided by 4 becomes n^2 + n +1/4.

    Factoring n, tn = n(n+1) + 1/4

    An odd number multiplied by itself will result in another odd number. adding two odd numbers will result in an even number.

    Therefore, for odd integers, n(n +1) + 1/4 is an even number with a remainder of 1/4
     
    Last edited: Apr 29, 2007
  7. Apr 30, 2007 #6

    HallsofIvy

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    No, the remainder is NOT 1/4!!

    Remember what you said initially:
     
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