Consider the questions of type :
Does $x$ divide $y$ ?
What are the most used divisibility theorems to decide such matters ?
Fermat’s little , quadratic and cubic reciprocity and $gcd(x,y) = gcd(x,x-y)$ are probably the most used.
What are the next most used or most useful ones ?
Btw I consider integers only here , So No gaussian integers or ring extensions.