Another way to say that 0|0 is that 0/0=an integer, and 0/0 is ... i'll say undetermined...
You can find out a counter example.I am not sure how to apply the transitive rules on this where if xPy and yPz then xPz.
The relation here is :
If xPy <=> p|gcd(x,y), with p prime
and yPz <=> p'|gcd(y,z), with p' prime
Does p'', prime, exist such as p''|gcd(x,z) ?
Counter example :
3P6 because 3, prime, divides 3 and 6.
6P14 because 2, prime, divides 6 and 14.
Does a prime number divide 3 and 14 ? The answer is no, because they are coprime (1 is not a prime number !)