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Math Help - Number theory

  1. #1
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    Number theory problems

    a number which sum of digit is 1977. Prove that the number will not be a perfect square
    Last edited by Swarnav; June 1st 2012 at 09:55 AM.
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  2. #2
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    Re: Number theory problems

    Hello, Swarnav!

    A number whose sum of digits is 1977.
    Prove that the number will not be a perfect square.

    Note: in this problem, a "number" means a positive integer.

    Consider the digital root of a number.
    Add the digits together.
    If the sum has more than one digit, add its digits together.
    Continue until we have a one-digit sum.
    That is the digital root of the number.

    A number can have only nine possible digital roots.
    Examine the digital roots of their squares.

    . . \begin{array}{ccc}\text{digit root} && \text{digital root} \\ \text{of }n & n^2 & \text{of }n^2 \\ \hline 1 & 1 & 1 \\ 2 & 4 & 4 \\ 3 & 9 & 9 \\ 4 & 16 & 7 \\ 5 & 25 & 7 \\ 6 & 36 & 9 \\ 7 & 49 & 4 \\ 8 & 64 & 1 \\ 9 & 81 & 9 \end{array}


    The digital root of a square must be: 1, 4, 7,\text{ or }9.


    Since the digital root of 1977 is 6, it cannot be a square.
    Thanks from Swarnav, linalg123 and Goku
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  3. #3
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    Re: Number theory problems

    but 6660 is not a perfect square, but it digit sum is 6+6+6+0=18,then 1+8=9.
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  4. #4
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    Re: Number theory problems

    Hello, Swarnav!

    But 6660 is not a perfect square, but it digit sum is 6 + 6 + 6 + 0 = 18, then 1 + 8 = 9.
    What is your point?


    Read my post again . . .

    I said: If a number is a square, then its digital root is 1, 4, 7 or 9.


    You are taking the converse:
    . . "If a number's digital root is 1, 4, 7 or 9, then it must be a square."
    I didn't say that.


    We cannot trust the converse.

    Consider the true statement: "If the figure is a square, then it has four side."

    The converse is: "If the figure has four sides, then it is a square."

    This statement may or may not be true . . . agreed?
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  5. #5
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    Re: Number theory problems

    yes,i have understood. thanks. but please can you help me to find a general form for all valid numbers?
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