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    I'm pondering partaking in the positivity posting pandemonium

    So I'm posting some motivational posters! :woo:

    Edit: click on the images to get full size. I forget how awful TSR is with images. I uploaded them all when I made the thread, but apparently you need to do them a maximum of one at a time per edit...




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    Why is the sum of the digits of any multiple of 9 also add up to a multiple of 9, Mr Magic Maths Man.
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    oh my grammar
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    (Original post by Melancholy)
    Why is the sum of the digits of any multiple of 9 also add up to a multiple of 9, Mr Magic Maths Man.
    A whole positive number has the form An(10^n) + An-1(10^(n-1)) + ... + A1(10) + A0 where An, An-1,..., A0 are between 0 and 9 (they are the digits of the number; this is just the normal decimal representation)

    This can be rewritten as

    An(10^n - 1) + An + An-1(10^(n-1) - 1) + An-1 + ... + A1(10 - 1) + A1 + A0
    = An(10^n - 1) + An-1(10^(n-1) - 1) + ... + A1(10 - 1) + A0 + A1 + ... + An

    It can be shown by induction that 10^k - 1 is divisible by 9 for any whole number k. Clearly 10^1 - 1 = 9 is, and if 10^k - 1 is then 10^(k+1) -1 = 10(10^k) - 1 = 10(10^k - 1) + 10 - 1 = 10(10^k - 1) + 9 which is clearly divisible by 9 since both the numbers being added are

    So we see then that everything before the A0 + A1 + ... + An in the above sum is divisible by 9. So the whole number is divisible by 9 if and only if A0 + A1 + ... + An is. This is just the sum of the digits.
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    Don't lie m8.
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    (Original post by Melancholy)
    Why is the sum of the digits of any multiple of 9 also add up to a multiple of 9, Mr Magic Maths Man.
    Just use modular arithmetic
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    Just used it and it didn't work.
 
 
 
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