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Math Help - A question from Number Theory book of Thomas Koshy?

  1. #1
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    A question from Number Theory book of Thomas Koshy?

    I have a question here from the book I have mentioned above

    The question is - There are 4 nos between 100 and 1000, which are equal to the sum of the cubes of its digits. Three of them are 153, 371 and 407. Find the fourth no.

    The answer is 370.

    I still actually didn't get as to how to proceed to solve the problem. Can anyone explain the solution to me.

    I found a pattern here - In each of the nos, the transition of the nos from one to the next, results in the change of only one digit

    153
    } 7
    371
    } 1
    370
    } 4
    407

    But I need a more plausible explanation as to how I can go about solving this and not use mere brute force!
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  2. #2
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    Quote Originally Posted by scw_0611 View Post
    I have a question here from the book I have mentioned above

    The question is - There are 4 nos between 100 and 1000, which are equal to the sum of the cubes of its digits. Three of them are 153, 371 and 407. Find the fourth no.

    The answer is 370.

    I still actually didn't get as to how to proceed to solve the problem. Can anyone explain the solution to me.

    I found a pattern here - In each of the nos, the transition of the nos from one to the next, results in the change of only one digit

    153
    } 7
    371
    } 1
    370
    } 4
    407

    But I need a more plausible explanation as to how I can go about solving this and not use mere brute force!
    I can't help you show there are 4 solutions, or even solve for the other 3, but I can show how to generate the last member if we are given one of the solutions.

    Call the number "xyz" = 100x + 10y + z. Then you require that
    100x + 10y + z = x^3 + y^3 + z^3

    Or
    (x^3 - 100x) + (y^3 - 10y) + (z^3 - z) = 0

    x(x^2 - 100) + y(y^2 - 10) + z(z^2 - 1) = 0

    Specifically what I'm after is the last term:
    x(x^2 - 100) + y(y^2 - 10) + z(z + 1)(z - 1) = 0

    If we have as a solution the number 371, note that z(z + 1)(z - 1) = 0 since z = 1. Thus we know that
    x(x^2 - 100) + y(y^2 - 10) = 0
    for this number.

    There is another way to make this last term 0: set z = 0. So 370 should also be a solution.

    -Dan
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  3. #3
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    Thanks a lot. Got it.
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