Prove that if the intended recipient takes the output and adds the key to it again $($using modular arithmetic$),$ then they will get the original message again. $($Hint: what happens if you add the key to itself?$)$ This process is known as decryption.

Why is it not possible to determine what the message is without knowing what the key is$?($Hint: suppose that you are given some output. Show that by changing the key, you can decrypt this output to any possible message.$)$