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You can compute the Smith normal form $B$ of the matrix with rows $a_1,\dots,a_n$. The rows generate $\mathbb{Z}^k$ if and only if the main diagonal entries of $B$ are $1$. (SNF over $\mathbb{Z}$ is determined only up to multiplication of the diagonal entries by $\pm 1$, and conventionally they are chosen to be nonnegative.)