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The key, it seems to me, is this: Joel Send a noteboard - 27/01/2012 08:54:23 PM
Essentially x0-x, where x is just a digit, not a number, will equal a multiple of 9.

That is sort of it, but going the wrong direction. 10x+x=11x; 11x-(x+x)=9x, by definition a multiple of 9. All other numbers between 10 and 99 are either:

10x+x+y, in which case 11x+y-(x+x+y)=9x, still a multiple of 9, or

10(x+y)+x, in which case 10x+10y+x-(x+x+y)=9x+9y, still a multiple of 9.
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