I'm beginning to study lattices and cannot solve the following exercise from the book:
Let $\Lambda=\left\langle\mathbb{Z}^n,\left(\frac{a_1}q,\ldots,\frac{a_n}q\right)\right\rangle$, where $\left(a_1,\ldots,a_n,q\right)=1$. Then $\det\Lambda=\frac1q$.
I can prove this in the case when each fraction can be reduced to the form $\frac{a_i'}{q_i}$ with $q_i$ pairwise relatively prime (using chinese remainder theorem), but the technique doesn't work in the general case.