number theory 1 x51 mod11 2how can we find the solution of

number theory !

1. x^5=1 mod11 ?

2.how can we find the solution of form of \' x^n=a \' modp ?

i wonder these two question from solving .. x^5+10=0 mod 11^4

Solution

1.x^5=1 mod11 ?

A: since 1 mod11 =1

x^5=1

x^5=1^5

x=1

2.

Let gcd(a,p)=1 where p is prime, and let n>0. Prove that the congruence equation x^n=a (mod p) has a solution if and only if ordp(a)|((p-1)/gcd(n,p-1))

Here, ordp(a) denotes the order of [a] in the field (F_p)*

What I have so far: Since p is prime, the group G=[Z/pZ)* is cyclic. Therefore, G^n={x^n : x is in G} = = where k=gcd(m,n) where m is order of G. In this case, m=p-1 so k=gcd(p-1,n)