00:01
Hi, today we are solving the question in which we need to find the primitive roots of modulo 34.
00:09
To do this we need to find the smallest positive integer k such that a raised to the power k is congruent to 1 mod 34 for all integers a relatively prime to 34.
00:40
Now the euler's question function phi of 34 is equals to 34 into 1 minus 1 by 2 into 1 minus 1 by 17 is equals to 16.
00:55
So we need to find the smallest k that divides 16.
01:00
Now let's find the divisor of 16.
01:03
So its divisor will be 1 2 4 8 and 16.
01:08
We will test each of these divisors to see if there exists an integer k a such that a raised to the power a k is not congruent to 1 mod 34.
01:23
So for k is equals to 1 we choose a is equals to 3.
01:30
Since 3 raised to the power 1 is not congruent to 3 that is not equal to 1 mod 34.
01:41
Similarly for k is equals to 2 a is equals to 3.
01:48
So since 3 square is equals to 9 that is not equivalent to 1 mod 34.
01:59
Now for k is equals to 4 we can take a is equals to 5.
02:08
Since 5 raised to the power 4 is equals to 625 that is not identical to 3 and it is not identical to mod 1 mod 34.
02:26
So for 8 we can choose a is equals to 3.
02:32
So it is also not identical.
02:34
Now if we take for k is equals to 8 and a is equals to 3...