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Math Help - congruence

  1. #1
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    congruence

    The sum of the digits of the number 8215 is 8+2+1+5=16 congruent to 7 (mod)9. Observe also that 8215=9(912) +7 congruent to 7 (mod)9. Does this hold for any number?Is the congruence class of any integer (mod 9) the same as the sum of its digits (mod) 9?
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  2. #2
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    Yes. To sum it up, every integer is congruent to the sum of their digits modulo 9. With congruences, this ends up being pretty easy to prove.

    Let n be some integer with the representation of its digits being: d_md_{m-1}d_{m-2}\cdots d_1d_0

    Note two things:
    (1) \color{red} 10 \equiv 1 \ (\text{mod }9)
    (2) n can be written as a sum of the powers of 10 (as with any other integer): n = d_m10^m + d_{m-1}10^{m-1} + d_{m-2}10^{m-2} + \cdots + d_110^1 + d_010^0

    Now we apply congruences:
      \begin{aligned}n & \equiv d_m{\color{red}10}^m + d_{m-1}{\color{red}10}^{m-1} + d_{m-2}{\color{red}10}^{m-2} + \cdots + d_1{\color{red}10}^1 + d_0{\color{red}10}^0  & (\text{mod } 9) \\ & \equiv \qquad \cdots \qquad \cdots \qquad \cdots \qquad \cdots \qquad \cdots \qquad \cdots \qquad \cdots   & (\text{mod } 9) \end{aligned}

    Can you figure it out from here?
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  3. #3
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    Appreciate the help

    I am not one of those people for whom math comes easy. When the prof. is writing these things in the board, I think I have it - only to get befuddled when I try to work on my own.
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