- #1

Shackleford

- 1,656

- 2

## Homework Statement

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.

## Homework Equations

The multiplicative order of

*a*modulo

*n*, denoted by ord

_{n}(

*a*), is the smallest integer k > 0 such that

*a*

^{k}≡ 1 (mod

*n*), when gcd(a,n) = 1 and n > 1.

## The Attempt at a Solution

1) I want to find an integer a such that

a

^{2}≡ 1 (mod 19),

*a*= 18;

a

^{3}≡ 1 (mod 19),

*a*= 7.

2) I'm looking at ord

_{17}(

*a*) = 4.

Well, two is a primitive root modulo 17, so

ord

_{17}(2

^{j}) = ord

_{17}(2)/gcd(8, j) = 8/ gcd(8, j) which implies that j = 6.

2

^{6}≡ 13 (mod 17), so a = 30 + 17k.