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Math Help - Sum of squares of digits dividing the number with those digits

  1. #1
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    Sum of squares of digits dividing the number with those digits

    How many numbers between 1 and 100 (excluded) satisfy that the sum of the squares of its digits divides the number?

    Help please


    Thank you.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by guidol92 View Post
    How many numbers between 1 and 100 (excluded) satisfy that the sum of the squares of its digits divides the number?

    Help please


    Thank you.
    Well clearly no one digit number greater than one satisfies this, so let's only look at two digit numbers.

    Let  n=10x+y where  0\leq x,y \leq 9

    We need to solve the Diophantine Equation  x^2+y^2 = 10x+y . Before I dive right in, let's see if you can do this.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by guidol92 View Post
    How many numbers between 1 and 100 (excluded) satisfy that the sum of the squares of its digits divides the number?

    Help please


    Thank you.
    Well clearly no one digit number greater than one satisfies this, so let's only look at two digit numbers.

    Let  n=10x+y where  0\leq x,y \leq 9

    We need to solve the Diophantine Equation  \displaystyle x^2+y^2 = 10x+y . Before I dive right in, let's see if you can do this.
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  4. #4
    Senior Member Shanks's Avatar
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    Quote Originally Posted by chiph588@ View Post
    Well clearly no one digit number greater than one satisfies this, so let's only look at two digit numbers.

    Let  n=10x+y where  0\leq x,y \leq 9

    We need to solve the Diophantine Equation  \displaystyle x^2+y^2 = 10x+y . Before I dive right in, let's see if you can do this.
    Do you notice the word "divides", not "equal to" in x^2+y^2 = 10x+y.
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Shanks View Post
    Do you notice the word "divides", not "equal to" in x^2+y^2 = 10x+y.
    Pardon?
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  6. #6
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    A computer search finds the numbers: 1, 10, 20 and 50.
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  7. #7
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    @Chip
    x^2 + y^2 | 10x + y
    so
    (x^2 + y^2)k = 10x + y
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  8. #8
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by elemental View Post
    @Chip
    x^2 + y^2 | 10x + y
    so
    (x^2 + y^2)k = 10x + y
    Ohh, I read the question wrong as Shanks was pointing out.
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