# Number Theory (2)

1. Jun 24, 2015

### Shackleford

1. The problem statement, all variables and given/known data

1. Find an integer modulo 19 with each of the following orders of 2 and 3.

2. Find all integers modulo 17 such that its order modulo 17 is 4.

2. Relevant equations

The multiplicative order of a modulo n, denoted by ordn(a), is the smallest integer k > 0 such that ak ≡ 1 (mod n), when gcd(a,n) = 1 and n > 1.

3. The attempt at a solution

1) I want to find an integer a such that

a2 ≡ 1 (mod 19), a = 18;

a3 ≡ 1 (mod 19), a = 7.

2) I'm looking at ord17(a) = 4.

Well, two is a primitive root modulo 17, so

ord17(2j) = ord17(2)/gcd(8, j) = 8/ gcd(8, j) which implies that j = 6.

26 ≡ 13 (mod 17), so a = 30 + 17k.

2. Jun 24, 2015

### Dick

I don't think you've actually asked a question. But a=4 works as well, that's not of the form 30+17k. Try to find where you missed that one.

Last edited: Jun 24, 2015
3. Jun 24, 2015

### Shackleford

Sorry. I just wanted to make sure that I was construing the questions correctly. I'll take a look.

Ah. I forgot gcd(8, j = 2) = 2.

Last edited: Jun 24, 2015