# Coprime numbers

Why is it true that if $a, b$ are coprime then there exists an $N$ such that for all $n> N,$ $n$ can be expressed as $n= ca+db$ where $c, d$ are non-negative?

Thanks.

-

By Bézout's identity there exists $p,q \in \mathbb{Z}$ such that $1 = pa + qb$ so we write $n = np a + nq b.$
We have $n = (np+kb)a + (nq -ka)b$ for all $k\in \mathbb{Z}.$
For both these coefficients to be positive we need $\displaystyle - \frac{np}{b} < k < \frac{nq}{a}$ and for sufficiently large $n$ this interval is large enough so that we can pick an integer from that range.