Proof Writing & Problem Solving · Superchallenge problem

A proof shows that no integer of the form $8k + 7$ is a sum of three squares, by checking the squares modulo $8$. A key lemma in a longer proof then says: if $4 \mid x^2 + y^2 + z^2$, then $x, y, z$ are all even. Using these, what is the smallest positive multiple of $4$ that is NOT a sum of three squares of integers?