Solve the congruence x^20 - 1 -= 0(mod 61)

Question
Congruence
asked 2021-01-06
Solve the congruence \(\displaystyle{x}^{{20}}-{1}\equiv{0}{\left(\text{mod}{61}\right)}\)

Answers (1)

2021-01-07
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) \(\displaystyle{x}\equiv{60}{\left(\text{mod}{61}\right)}\Rightarrow{x}={60}+{61}{l},{l}\in\mathbb{Z}\)
0

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