The following lemma is not hard to prove.

**Lemma** : Let $c_1 \neq c_2 \neq \dots \neq c_r \in [n]$ and $k \in [n]$. If $m_1, m_2, \dots, m_r$ are integers (some of them might be negative) such that $m_1c_1 + m_2c_2 + \dots + m_rc_r = k$, then $\exists$ integers $m'_1, m'_2, \dots,m'_r$ satisfying $m'_1c_1 + m'_2c_2 + \dots + m'_rc_r = k$ such that $|m'_1|+|m'_2|+\dots+|m'_r| \leq poly(n)$. Here $poly(n)$ means $O(n^c)$ for some positive constant $c$.

I am guessing that the above lemma is well known. I am looking for a reference of the above lemma and the best possible bound for $poly(n)$.