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Find all solutions of the following congruence 3x^5≡1(mod 23)

This is what I have so far

I know 5 is a primitive root and I made a table of indices modulo 23 with respect to 5

then

Φ(23)=22

Ind5(3x5)≡ind5(1)=22(mod 22)

Ind5(3x5)≡ind5(3) + ind5(x5)≡16 + 5ind5(x)(mod22)

16+5ind5(x)≡22(mod22)

5ind5(x)≡6(mod22)

I'm stuck here because I'm not really sure how to get rid of the 5 on the left side or if I even have to.

Thanks

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# Homework Help: Number theory help

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