The answer is yes. In general: Let \(\displaystyle a\) and \(\displaystyle b\) be integers. There exists two integers \(\displaystyle x\) and \(\displaystyle y\) such that \(\displaystyle ax+by=1\) if and only if \(\displaystyle (a,b)=1\). In your case, \(\displaystyle a=n\), \(\displaystyle x=-1\) and \(\displaystyle b=n

+1\), \(\displaystyle y=1\).