Using much of what was discussed in my last talk, we will prove a very important theorem that deals with a prime expansion of any rational number. An understanding of the following concepts are necessary to complete the proof:

Relatively Prime
Two positive integers m and n are relatively prime if and only if there exist integers s and t such that ms + nt = 1.

Division Algorithm
a = bq + r

Partial Fractions in Calculus Number Theory, and Algebra,C. A. Yackel and J. K. Denny (Vol 38, No. 5, November 2007 The College Mathematics Journal)

In the article, the Theorem we will prove is Theorem 2.