Problem

Source: Romania NMO - 2018

Tags: greatest common divisor, vector, linear algebra, matrix



Let n2 be a positive integer and, for all vectors with integer entries X=(x1x2xn)let δ(X)0 be the greatest common divisor of x1,x2,,xn. Also, consider AMn(Z). Prove that the following statements are equivalent: i)  |det \textbf{ii) } \delta(AX)=\delta(X), for all vectors X \in \mathcal{M}_{n,1}(\mathbb{Z}). Romeo Raicu