Let $r$ and $r'$ be coprime positive integers, and let Consider integers $z$ such that $1 \leq z \leq rr'$ and $\gcd(z, rr') = 1$.
I want to prove that there is a one to one corrispondance between each integer $z$with the above properties and a a pair $x,y$ of integers such that $1 \leq x \leq r$ , $1 \leq y \leq r'$, $\gcd(x, r) = 1$, $\gcd(y, r) = 1$.
Moreover prove that $$\frac{z}{rr'} - \frac{x}{r} - \frac{y}{r'} \in \mathbb{Z}$$
My try:
The first part should follow from the Chinese Remainder Theorem (CRT), the residue class of $z \pmod{rr'}$ uniquely determines residue classes $x \pmod{r}$ and $y \pmod{r'}$ such that:
$$z \equiv x \pmod{r}, \quad z \equiv y \pmod{r'}.$$
But I am not sure how that can be expressed in terms of the gcd.
For the second part
The CRT states that $z$ can be expressed as:
$$z = r m + x = r' m' + y,$$
where $m, m'$ are integers. Dividing through by $rr'$, we have:
$$\frac{z}{rr'} = \frac{x}{r} + \frac{m}{r'} = \frac{y}{r'} + \frac{m'}{r}.$$
But I haven't been able to get anything meaningful from this
Alternatively I tried using the explicit construction of $z $ in terms of $x,y,r$ and $r'$ that one obtains when proving the existence of the solution of the system of congruences in the CRT, as stated for example in wikipedia (sometimes called the explicit CRT) that is $z \equiv xr'm+yrn ($mod $rr')$, with $m$ and $n$ integers, but also this does not help. Any help would be appreciated
