Step 1

The given congruence

\(\displaystyle{x}^{{20}}-{1}\equiv{0}{\left(\text{mod}{61}\right)}\)

Step 2

\(\displaystyle{x}^{{20}}\equiv{1}{\left(\text{mod}{61}\right)}\)

\(\displaystyle{x}\equiv{1}{\left(\text{mod}{61}\right)}\)

\(\displaystyle{x}\equiv-{1}\equiv{60}{\left(\text{mod}{61}\right)}\)

1) \(\displaystyle{x}\equiv{1}{\left(\text{mod}{61}\right)}{x}={1}+{61}{k},{k}\in\mathbb{Z}\)

2) \(\displaystyle{x}\equiv{60}{\left(\text{mod}{61}\right)}\Rightarrow{x}={60}+{61}{l},{l}\in\mathbb{Z}\)

The given congruence

\(\displaystyle{x}^{{20}}-{1}\equiv{0}{\left(\text{mod}{61}\right)}\)

Step 2

\(\displaystyle{x}^{{20}}\equiv{1}{\left(\text{mod}{61}\right)}\)

\(\displaystyle{x}\equiv{1}{\left(\text{mod}{61}\right)}\)

\(\displaystyle{x}\equiv-{1}\equiv{60}{\left(\text{mod}{61}\right)}\)

1) \(\displaystyle{x}\equiv{1}{\left(\text{mod}{61}\right)}{x}={1}+{61}{k},{k}\in\mathbb{Z}\)

2) \(\displaystyle{x}\equiv{60}{\left(\text{mod}{61}\right)}\Rightarrow{x}={60}+{61}{l},{l}\in\mathbb{Z}\)