00:01
In this problem, we're given that 13 is a primitive root mod 31.
00:05
So the prime p in this problem is 31.
00:08
Our first task is to find all of the primitive roots, mod 31.
00:12
The second task is to find all the elements of order 15.
00:15
And finally, the third task is to find all elements of order 5.
00:20
So the key to all three parts is to note that 13 generates all of the numbers from 1 through 30 mod 31.
00:28
And so we can look at the powers of 13.
00:30
For example, in part a, we want the order to be 30, so that these will be primitive roots.
00:39
So we want every number.
00:43
The primitive root will consist of all of the numbers that are powers of 13.
00:50
So we have 13 to the k.
00:53
We want to take the order that we want and divide that into 30.
00:56
In this case, we get 1.
00:59
So we want values of k where the gcd of k with 30 equals 1.
01:05
So this will describe all of the primitive roots, mod 31.
01:11
And it turns out there are eight such exponents.
01:14
We can check that by taking the euler fee function of 30.
01:17
Since 30 is 2 times 3 times 5, its oiler fee value is 1 times 2 times 4 or 8.
01:24
So the primitive roots can be written in this way.
01:27
13 to the 1st, 13 to the 7th, 13 to the 11th, 13 to the 13th to the 13th, 13th to the 13th, 13 to the 17th, 13th, and we start to see some patterns with those exponents.
01:43
If k works, then so does 30 minus k.
01:46
So we have 13 to the 19 there, and we have 13 to the 23, and finally 13 to the 29.
01:54
Now all we have to do is reduce these mod 31.
01:58
So here's our list, reducing each of these powers in turn.
02:01
Modulo 31.
02:04
We have 13, 22, 3, 11, 17, 17.
02:17
21, 24, and finally 12.
02:22
So here are all eight of the primitive roots, mod 31.
02:30
So let's use a similar strategy for part b...