Proving x^6 ≡ 1 (mod 7) Using Fermat's Little Theorem

[halvizo1031]

Homework Statement

I'm suppose to prove that if (x,7)=1, then x to the 6th is congruent to 1 mod 7.

Homework Equations

The Attempt at a Solution

Now, i have the proof by induction when (a,p)=1 but how do i apply this to prove it when a=x and p=7?

 

[VeeEight]

What does Fermat's Little Theorem state?

 

[halvizo1031]

it states that if (a,p)=1 then a^(p-1) is congruent to 1 (mod p)

 

[VeeEight]

7 is a prime, and you have that (x,7), so use Fermat's Little Theorem on x^7-1 = x⁶

 

[halvizo1031]

so how is this for an answer?:

since 7 is a prime and the gcd(x,7) =1, then by Fermat's Little Theorem,

x^(7-1)=x^6 is congruent to 1(mod7)

 

[VeeEight]

Yes.

 

[halvizo1031]

so now, if I want to show that (x^3)^2 is congruent to +/-1 (mod 7) would my work be correct? (please see the attachment).