Homework 4: Solutions
Section 2.8: 1, 4, 9
1.
Find a primitive root of the prime 3; the prime 5; the prime 7; the
prime 11; the prime 13.
3:
2 (this is the only primitive root
mod
3 as
φ
(3

1) =
φ
(2) = 1)
5:
2 and 3 are primitive roots (again, notice that
φ
(5

1) =
φ
(4) = 2,
so these are the only ones)
7:
3 and 5 are primitive roots (
φ
(7

1) =
φ
(6) = 2, thus there are no others)
11:
Notice that 11

1 = 10 = 2
·
5. We know that if an element
a
has order
h
,
and (
a,
11) = 1, then
h

φ
(11), i.e.
h

10. Since (2,11)=1, the order of 2 is 1, 2,
5, or 10. Since 2
1
≡
2(
mod
11), 2
2
≡
4(
mod
11), and 2
5
= 32
≡ 
1(
mod
11),
it must be that 2
10
≡
1(
mod
11). All in all, 11 has 4 primitive roots: 2, 6, 7, 8
13:
Notice 13

1 = 12 = 2
2
·
3, so there are 4 primitive roots. 2
6
≡

1(
mod
13), and 2
4
≡
3(
mod
13), 2
3
≡
8(
mod
13), 2
2
≡
4(
mod
13), so the
order of 2 must be 12. The primitive roots of 13 are: 2, 6, 7, 11
...