Is $\mathbb{Z}$/(10) isomorphic to $\mathbb{Z}$/(2) $\times$ $ \ \mathbb{Z}$/(5)? a) Is $\mathbb{Z}$/(10) isomorphic to $\mathbb{Z}$/(2) $\times$  $ \ \mathbb{Z}$/(5)?
b) Is $\mathbb{Z}$/(8) isomorphic to $\mathbb{Z}$/(2) $\times$  $ \ \mathbb{Z}$/(4)?
I'm not sure how I would do these questions? 

For the first one you can consider the ring homomorphism
$$\varphi:\Bbb Z_{10}\to\Bbb Z_2\times\Bbb Z_5\\
k\mapsto(k\%2,k\%5)$$
and show that it is an isomorphism. Remember: Both sides have $10$ elements, so it suffices to either show injectivity or surjectivity to get the bijectivity. 
The second case is not even true for groups. Can you find an element in $(\Bbb Z_8,+)$ of an order $n$ such that no element in the left side has order $n$ ?
