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Math Help - yr 8 homework

  1. #1
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    yr 8 homework

    Find all the three-digit even numbers N such that 693 x N is a perfect square, that is 693 x N = k squared whre k is an integer.

    I know that 308 is oneof them but how do we no that it is the only one? are there more?

    I think I bascaly kno but I just want to be sure, please explain carefully
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by mathswonderer View Post
    Find all the three-digit even numbers N such that 693 x N is a perfect square, that is 693 x N = k squared whre k is an integer.

    I know that 308 is oneof them but how do we no that it is the only one? are there more?

    I think I bascaly kno but I just want to be sure, please explain carefully
    first, get the factorization of 693:

    693 = (3^2)(7)(11)

    now, start finding N such that it has factors 7 and 11 whose powers are odd, and other integer factors whose powers are even.

    for example: N= (7)(11)(2^2) = 308 and you are correct..

    N= (7)(11)(3^2) = 693
    N= (7)(11)(4^2) = ?.. just make sure that N is still a 3 digit number..

    EDIT: basically, i chose 1 as the powers of 7 and 11, since it it were 3 or higher odd powers, then it will give you not a three digit number.. EX: (7^3)(11) = 3773
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  3. #3
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    I understand that but how do we know that 308 is the only one, how can I simtimatically show that it is the onnly one?
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  4. #4
    MHF Contributor kalagota's Avatar
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    308 is not the only one..

    note that 693 is also a 3-digit number, and if you multiply it to itself, it will give you a perfect number,,

    have you solve the last N? is it a 3-digit number? if it is, then you got another number other that 308 and 693.. and if not, then there are no other numbers..
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  5. #5
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    SO would this working be sufficient?
    1. 7 x 11 x 2= 154
    693 x 154=106 722
    Sqrt=326.683332

    2. 7 x 11 x 22=308
    693 x 308 = 213 444
    Sqrt=462

    3. 7 x 11 x 23=616
    693 x 616=426 888
    Sqrt=653.36666

    4. 7 x 11 x 24=1232
    693 x 1232= 853 776
    Sqrt=924
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  6. #6
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by mathswonderer View Post
    SO would this working be sufficient?
    1. 7 x 11 x 2= 154
    693 x 154=106 722
    Sqrt=326.683332

    2. 7 x 11 x 2x2=308
    693 x 308 = 213 444
    Sqrt=462

    3. 7 x 11 x 2x3=616
    693 x 616=426 888
    Sqrt=653.36666

    4. 7 x 11 x 2x4=1232
    693 x 1232= 853 776
    Sqrt=924
    1. NO! since the square root is not an integer..
    2. yes!
    3. NO! same reason as 1.
    4. NO! since 1232 is not a 3-digit number..

    besides, for number 3: 7 x 11 x 2 x 3 = 462
    for number 4: 7 x 11 x 2 x 4 = 616
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