Understanding the functionality of groups, cyclic groups and subgroups is im$-$ portant for the use of public$-$key cryptosystems based on the discrete logarithm problem. That$$s why we are going to practice some arithmetic in such structures in this set of problems.

Let$$s start with an easy one. Determine the order of all elements of the multi$-$ plicative groups of:

$1.$ Z $52.$ Z $73.$ Z $13$

Create a list with two columns for every group, where each row contains an element a and the order ord$($a$).$

$($Hint: In order to get familiar with cyclic groups and their properties, it is a good idea to compute all orders $$by hand$,$ i$.$e$.,$ use only a pocket calculator. If you want to refresh your mental arithmetic skills, try not to use a calculator whenever possible, in particular for the first two groups.$)$