We have readers of all backgrounds here at Ars Math, so I thought I would experiment with a more expository post than usual. Commenter Wendell Dryden is teaching himself the Chinese remainder theorem.
The result has been considerably generalized (as the Wikipedia entry makes clear), and one variant is easier to understand (I think): polynomial interpolation. Given n numbers xi on the x-axis, and n numbers, yi, on the y-axis, you can always find a polynomial p such that p(xi) = yi. The steps in solving this problem and the integer congruence problem are similar, and both problems can be solved by using the extended Euclidean algorithm.
This analogy has been taken much further in algebraic geometry. From that point of view, an integer is no longer just a number, but actually (like a polynomial) a function. The integer, in its function guise, sends prime numbers to that integer modulo that prime. So the function 67 sends 2 to 1, 3 to 1, 5 to 2, 7 to 4, 11 to 1, etcetera. (The value eventually stabilizes, in this instance at 67, which always seemed to me must be a fact of some significance, but I’ve never seen it used for anything.)
So now you know: to an algebraic geometer, integers are functions. Mathematics is like drugs, but cheaper.