Prove that a cyclic group of order $\displaystyle n$ can be written as the direct product of its subgroups if $\displaystyle n$ is divisible by atleast 2 distinct primes.
Prove that a cyclic group of order $\displaystyle n$ can be written as the direct product of its subgroups if $\displaystyle n$ is divisible by atleast 2 distinct primes.
I am not sure what you mean be "direct product of its subgroups".
If $\displaystyle n=p_1^{a_1}\cdot ... \cdot p_k^{a_k}$ then $\displaystyle \mathbb{Z}_n = \mathbb{Z}_{p_1^{a_1}}\times ... \times \mathbb{Z}_{p_k^{a_k}}$