# Fermat's little theorem

1. Apr 25, 2010

### halvizo1031

1. The problem statement, all variables and given/known data
I'm suppose to prove that if (x,7)=1, then x to the 6th is congruent to 1 mod 7.

2. Relevant equations

3. 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?

2. Apr 25, 2010

### VeeEight

What does Fermat's Little Theorem state?

3. Apr 25, 2010

### halvizo1031

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

4. Apr 25, 2010

### VeeEight

7 is a prime, and you have that (x,7), so use Fermat's Little Theorem on x7-1 = x6

5. Apr 25, 2010

### 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)

6. Apr 25, 2010

### VeeEight

Yes.

7. Apr 25, 2010

### 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).

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