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

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

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

### Relevant Questions

Solve the congruence $$x^{20}-1\equiv 0(mod\ 61)$$
Using Fermat's Little Theorem, solve the congruence $$\displaystyle{2}\cdot{x}^{{{425}}}+{4}\cdot{x}^{{{108}}}-{3}\cdot{x}^{{2}}+{x}-{4}\equiv{0}\text{mod}{107}$$.
Write your answer as a set of congruence classes modulo 107, such as {1,2,3}.
Solve the linear congruence
$$\displaystyle{x}+{2}{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle{2}{x}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
Solve the congruence $$\displaystyle{x}^{{2}}\equiv{1}{\left(\text{mod}{105}\right)}$$
Using the Chinese Remainder Theorem, solve the following simultaneous congruence equations in x. Show all your working.
$$\displaystyle{9}{x}\equiv{3}\text{mod}{15}$$,
$$\displaystyle{5}{x}\equiv{7}\text{mod}{21}$$,
$$\displaystyle{7}{x}\equiv{4}\text{mod}{13}$$.
Determine whether the congruence is true or false.
$$100\equiv 20\ mod\ 8$$
Show that the congruence $$\displaystyle{8}{x}^{{2}}-{x}+{4}\equiv{0}{\left(\text{mod}{9}\right)}$$ has no solution by reducing to the form $$\displaystyle{y}^{{2}}\equiv{a}{\left(\text{mod}{9}\right)}$$
$$\displaystyle{7}{x}\equiv{13}{\left(\text{mod}{19}\right)}$$
$$\displaystyle{7}{x}+{3}{y}\equiv{10}{\left(\text{mod}{16}\right)}$$