Harristotelian Logic

My little congruence result is a way to highlight something I puzzle over when I figure out my mathematical results:

With non-zero coprime integers n_1 and n_2, if f_1 = r_1 mod n_1 and f_1 = r_2 mod n_2, you can find f_1 mod p_1*p_2 with

f_1 = r_1 + j*n_1 mod n_1*n_2

where j = (r_2 - r_1)*n_1^{-1} mod n_2.

Now that's easy. But does it work? Let's see. Let n_1 = 11 and n_2 = 23, and let r_1 = 4 and r_2 = 9, then

f_1 = 4 + 11j mod 11(23) and j = (9 - 4)*11^{-1} mod 23 = 5(21) mod 23 = 13 mod 23, so

f_1 = 4 + 11(13) mod 11(23) = 147 mod 253.

Easy. 147 mod 11 = 4 and 147 mod 23 = 9, as required.

Now I discovered that partly because I'd read over the Chinese Remainder theorem and it just didn't grab me as something I liked all that much, so I just went and figured out an easy way to do the same thing with my own research, as, um, I like to use my own research!

So why is my little congruence result new?

And with it shown, why would the mathematical community just quietly not acknowledge it?

I puzzle over such things as I try to figure out how you people think.

What motivates you? What do you think you're doing when you do mathematics? What do you think math is?

What is important to any of you about mathematics? Where in it do you see value? And why?