...deduce a pattern. Is this correct? I'm learning deductive reasoning in math class where we're being taught to prove certain mathematical patterns logically. For example, adding an odd number and another odd number always equals an even number - using deduction, it is broken down to this equation:

(2x - 1) + (2y - 1) = 2n
2x + 2y - 2 = 2n
2(x + y - 1) = 2n

The above equation is divisible by 2; therefore, the answer logically has to be even.

Anyways, that got me to thinking. When you add the digits of any multiple of 9, the answer is itself a multiple of 9. I've never really understood why, it's something I just knew. But I figured there has to be a logical way to deduce that pattern as well. I've thought about it for a while, and this is what I came up with (I let the variables "x, y, z, etc." represent the digits, with "9t" as the multiple of 9):

1) 9t = x + 10y + 100z + ... *I added a power of ten each time as an indicator of place value*
2) 9t = x + (9y + y) + (99z + z) + ... *Here I tried to make the numerals beside the variable divisible by 9*
3) 9t = (x + y + z + ...) + (9y + 99z + ...)
4) 9t = (x + y + z + ...) + 9(y + 11z + ...) *I'll represent the sum of the variables (ie. the digits) in the first bracket as "s" and in the second bracket as "a"*
5) 9t = s + 9a
6) 9t - 9a = s
7) 9(t - a) = s

The equation is divisible by 9; therefore, "s" (the sum of the digits) has to be a multiple of 9.

This isn't a question from my homework or anything, I'm just posting it here to have someone more experienced in mathematics look over my work and tell me if my reasoning is flawed. Thanks! =)