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Math Help - Find all three digit even numbers N such that 693 times N is a perfect square.

  1. #1
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    Exclamation Find all three digit even numbers N such that 693 times N is a perfect square.

    Find all three digit even numbers N such that 693 times N is a perfect square.

    I have no idea how to go about this?

    Please help urgently!
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  2. #2
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    Quote Originally Posted by vonge View Post
    Find all three digit even numbers N such that 693 times N is a perfect square.

    I have no idea how to go about this?

    Please help urgently!
    First of all observe that 693 = 9 .11 . 7. So if 693N is a perfect square, then 77|N since only one factor of 7 and 11. More clearly N = 77k^2.

    Now choose even k such that 77k^2 is a three digit number

    You will see that for k= 4, N is already a four digit number. Thus k=2 is the only answer.

    Or N = 77 (2)^2 = 308

    So the only three digit that satisfies is N = 308
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    Thank you so much for that
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