lwfrgin
2021-02-02
Answered

Determine the number of solutions of the congruence $x4\equiv 61\left(\text{mod}117\right)$ .

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Caren

Answered 2021-02-03
Author has **96** answers

Step 1

Given:${x}^{4}\equiv 61\left(\text{mod}117\right)$

$117={3}^{2}\times 13$

As$\frac{\varphi \left(9\right)}{4,\varphi \left(9\right)}=\frac{6}{4,6}=\frac{6}{2}=3$ (here(4,6) denotes the g.c.d of(4,6))

and${\left(61\right)}^{3}\equiv {(-2)}^{3}\equiv 1\left(\text{mod}9\right)$

we deduce the congruence

${x}^{4}\equiv 61\left(\text{mod}9\right)has(4,\varphi \left(9\right))=(4,6)=2$ solutions

Step 2

Similarity$\frac{\varphi \left(13\right)}{4,\varphi \left(13\right)}=\frac{12}{4,12}=\frac{12}{4}=3$

and${\left(61\right)}^{3}\equiv {(-4)}^{3}\equiv 1\left(\text{mod}13\right)$

So, the congruence${x}^{4}\equiv 61\left(\text{mod}13\right)has(4,\varphi \left(13\right))=(4,12)=4$ solutions

hence, the number of solutions of the congruence${x}^{4}\equiv 61(\mp d117)is2\times 4=8$ .

Given:

As

and

we deduce the congruence

Step 2

Similarity

and

So, the congruence

hence, the number of solutions of the congruence

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