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# The Positivity Thread

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Why bother with a post grad? Are they even worth it? Have your say! 26-10-2016
1. I'm pondering partaking in the positivity posting pandemonium

So I'm posting some motivational posters!

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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2. Why is the sum of the digits of any multiple of 9 also add up to a multiple of 9, Mr Magic Maths Man.
3. oh my grammar
4. (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.
5. Don't lie m8.
6. (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
7. Just used it and it didn't work.

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