How would I go about finding the torsion coefficients of Z10 x Z36 x Z14 x Z21?
I think the first stage is to write 10, 36, 14 and 31 as products of primes but I'm not sure where to go from there.
I am not exactly sure what you are asking but I it seems to me an exercise in the classification theorem for abelian groups. You need to bring this to standard form. Note $\displaystyle \mathbb{Z}_{10} \simeq \mathbb{Z}_2\times \mathbb{Z}_5$, and so on. Now replace each factor with the equivalent isomorphic form to bring this expression into a direct product of groups of the form $\displaystyle \mathbb{Z}_{p^k}$.