tag removed - Strong Bezout's Identity?

Let ${ a_i }_{i=1}^N $ be a set of elements of the ring of integers, $mathbb{Z}_D$ and define $g = text{gcd}(a_1, a_2,ldots, a_N, D)$. Then Bezout's Identity states that there exists another set ${ x_i }_{i=1}^N $ such that

$sum_{i=1}^N a_i x_i equiv g bmod D$

For my work, I needed to show that such a solution set ${ x_i }_{i=1}^N $ exists with an ADDITIONAL requirement that $x_1$ must be coprime to $D$. I managed to prove this stronger version of Bezout's Identity using Chinese Remainder Representation (correctly I hope).

My question : Is this result well-known under another name? Do you know of any references discussing this result? Or is this a special case of an even stronger form of Bezout's Identity?