Proof that if [(a,b) = (c,d) = 1] and [a/b + c/d ∈ Z], then b = d

Since \(\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}\) is an integer, then \(bd | ad + cb\), which means:

$$\begin{gather} bd | d(ad + cb) \\ bd | ad^2 + cbd \end{gather}$$

Since \(bd | cbd\):

$$\begin{gather} bd | ad^2 \\ b|ad \\ (a,b)=1 \implies b|d \end{gather}$$

We can make a similar argument to show that \(d|b\), which means \(b=d\).

Proof That If [(a,b) = (c,d) = 1] And [a/b + c/d ∈ Z], Then b = d