Existence of quotient and remainder

Base case. If $n<d$: $q=0$, $r=n$, unique.

Inductive step. Suppose all $k<n$ have a unique decomposition. For $n$:
- If $n<d$, done.
- If $n\ge d$: $n-d<n$, so by induction $n-d=qd+r$ with $0\le r<d$. Then $n=(q+1)d+r$, giving existence.

Uniqueness. Suppose $n=qd+r=q'd+r'$. Then $(q-q')d=r'-r$ with $|r'-r|<d$; this forces $r=r'$ and $q=q'$.

Conclusion. Euclidean division exists and is unique.