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[polymath257]

What's the deal with 3, 6, and 9? (I think it's just about 3 but.........)

3x4=12, 1+2=3
6x24=144, 1+4+4=9
9x2025=18225, 1+8+2+2+5=18, 1+8=9
6x85069=510414, 5+1+0+4+1+4=15, 1+5=6
3x50647921=151943763, 1+5+1+9+4+3+7+6+3=39, 3+9=12, 1+2=3

Nonfunctional math guy is curious.

It boils down to the fact that if you divide 10 (the base of our number system) by 3 or 9, the remainder is 1. because of that, when you divide 100 or 10000 or any power of 10 by 3 or 9, the remainder will be 1.

This is actually the basis of a lot of strange arithmetic tidbits.

The effect is that when you divide a number by 3 or 9, the remainder will be the same as what you get if you divide the sum of the digits by 3 or 9. You can keep doing this until you have a single digit number.

So, if you start out with a multiple of 3, the final, single digit has to be 3, 6, or 9.

If you start out with a multiple of 9, the single digit has to be 9.

Here is a variant, that is due to the fact that 10 is the same as -1 when you divide by 11.

Take any multiple of 11 and alternately add and subtract its digits. Than do the same with the answer. Eventually, you will get to 0.

Example:

11x5971=65681. 6-5+6-8+8-1=11, 1-1=0.

11x821074=9031814, 9-0+3-1+8-1+4=22, 2-2=0