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

Solve the congruence ${x}^{20}-1\equiv 0\left(\text{mod}61\right)$
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Step 1
The given congruence
${x}^{20}-1\equiv 0\left(\text{mod}61\right)$
Step 2
${x}^{20}\equiv 1\left(\text{mod}61\right)$
$x\equiv 1\left(\text{mod}61\right)$
$x\equiv -1\equiv 60\left(\text{mod}61\right)$
1) $x\equiv 1\left(\text{mod}61\right)x=1+61k,k\in \mathbb{Z}$
2) $x\equiv 60\left(\text{mod}61\right)⇒x=60+61l,l\in \mathbb{Z}$