Let $f: R\to S$ be a homomorphism of rings (with $R$ commutative) such that kernel of $f$ has $4$ elements and image of $f$ has $16$ elements. How many elements does $R$ have?
Would you simply use the first isomorphism theorem in the following way? $R/\ker(f)=R/4$, so $R$ must have $16\cdot 4= 64$ elements.