I could use help on the following problem: Suppose $v_1, \dots, v_k$ are linearly independent in $\mathbf{Z}^n$. Then show that the cardinality of the torsion subgroup of $\mathbf{Z}^n/\langle v_1, \dots, v_k \rangle$ is exactly the largest integer dividing $v_1 \wedge \cdots \wedge v_k$.
Here's what I've tried: I know that we can assume $k \leq n$ (otherwise, we can lift the set to $\mathbf{Q}^n$, which is a vector space, and hence $k > n$ implies the set has a nontrivial $\mathbf{Q}$-linear dependence, which can be converted to a dependence over $\mathbf{Z}$ by multiplying by the product of the denominators). Hence, assuming $k \leq n$, we can consider the endomorphism $T : V \to V$, given by $e_i \mapsto v_i$ if $i \leq k$, and otherwise $e_i \mapsto 0$. This is a map given by a matrix with columns equal to $v_i$. The determinant of which is exactly the largest integer. I don't know how to connect this to the torsion subgroup part though! I could use any hints.