Chinese Remainder theorem: How is it used in this passage? I am trying to understand this passage in 'Number theory and geometry' by Álvaro Lozano-Robledo :

How is the Chinese Remainder theorem used here, exactly ? 
My understanding is that the $s$ and $t$ need to be relatively prime ie $\gcd(s,t)=1$ ? 
Also what is the meaning of $ P \mod s$ in this context where $P=(1/3,1/3)$ ?
 A: The CRT is used in the standard way.  Since $(3,m)=1$, there's an inverse for $3\pmod m$.  Hence in $\Bbb Z/m\Bbb Z$, $1/3$ makes sense.
You probably know that if $(3,t)=1$, then $3$ has an inverse mod $t$.  Say $t=2$.  Then $1/3=1$, since $1\cdot3\cong1\pmod2$.
For instance, in the case of $m=15$, it is necessary that $s=3$ and $t=5$.  So we can work everything out.
I recommend doing the suggested exercise in this case, to get a feel for what's going on.  
So, compute $1/3\pmod5$.  I get $2$.  So in $\Bbb Z/5\Bbb Z$, we have $P=(2,2)$.  Now do $Q$ in $\Bbb Z/3\Bbb Z$.  I get $Q=(0,2)$.
Now use CRT, to get $R$.    We get $R=(12,2)\in C(\Bbb Z/15\Bbb Z)$.
And we can check that $2\cdot12^2+7\cdot2^2\cong1\pmod{15}$.  
This procedure works for any $m\gt1$.  Thus we have an example of a diophantine equation that has solutions mod $m$ for every $m$ greater than $1$, but which has no integral solutions.  However, it has rational solutions.
The author has broken the problem into $3$ cases.  The key, which he didn't make very clear, is that in case $m$ is a multiple of $15$, it will be possible to write $m=s\cdot t$, with $s$ and $t$ relatively prime, as well as having $(s,5)=1=(t,3)$.  This is clear.  Just put $t=5^k$, where $k$ is the greatest power of $5$ that divides $m$.  This should resolve your issue.
