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Math Help - Show that pos int divisible by 3 iff sum of digits divisible by 3

  1. #1
    Member oldguynewstudent's Avatar
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    Show that pos int divisible by 3 iff sum of digits divisible by 3

    Show that a positive integer is divisible by 3 iff the sum of its digits is divisible by 3.

    This is from Rosen but I can't follow the proof in the solutions manual (again). Section 3.6 problem 29.

    Please explain the detail so I'll understand the whole process.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by oldguynewstudent View Post
    Show that a positive integer is divisible by 3 iff the sum of its digits is divisible by 3.

    This is from Rosen but I can't follow the proof in the solutions manual (again). Section 3.6 problem 29.

    Please explain the detail so I'll understand the whole process.
    What part of the proof eludes you?
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  3. #3
    Senior Member Sampras's Avatar
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    Suppose  n = a_0 + a_{1}10+ a_{2}10^{2} + \cdots + a_{n} 10^n .

    Consider  s = a_0 + \cdots a_n and look at n-s.
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  4. #4
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Drexel28 View Post
    What part of the proof eludes you?
    Since 10 \equiv 1 (mod 3) we have a \equiv a_n _- _1 + ... + a_1 + a_0 (mod 3). Therefore a \equiv 0 (mod 3) iff the sum of the digits is congruent to 0 (mod 3).

    I don't follow the part we have a \equiv sum (mod 3).

    Is this because 10 to any power divided by 3 has a 1 remainder? Then a divided by 3 leaves the sum of the digits?

    I just now saw this relationship. The solutions manual assumes this is intuitively obvious, I guess.
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  5. #5
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    Let x = \sum _{i=0} ^n a_i 10^i.

    Suppose 3 | x then 0 \equiv x \equiv \sum _{i=0} ^n a_i {10} ^n  \equiv \sum _{i=0} ^n a_i {1} ^n \bmod{3}
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    Re: Show that pos int divisible by 3 iff sum of digits divisible by 3

    Let a number be abc
    = 100a + 10b + c
    = 99a + 9b + a + b + c
    99a and 9 b are divisible by 3
    so (a + b + c) must be divisible by 3.
    so (a + b + c) is sum of digits must be divisible by 3.
    ---
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