# Primitive Roots

Gear300
Hello friends from afar.

I ran into what I felt to be somewhat of an odd question:

Prove that some odd numbers are primitive roots modulo pm for each odd prime p and each positive integer m.

It feels dodgy given that any odd number n = p1p2 ⋅⋅⋅ ps cannot be a primitive root of a prime number involved in its prime factorization. I just needed to be sure. Many thanks.

## Answers and Replies

Mentor
2021 Award
The wording is quite disturbing and I stumbled upon the same argument as you. "some odd numbers" looks strange.
It would make more sense the other way around (or I didn't get the point either):

For each odd prime ##p## and each positive integer ##m## prove that some odd numbers are primitive roots modulo ##p^m.##

Gear300
Indeed. I'm guessing yours is how it's done, since it seems like the original could be semantically interpreted like that. Thanks.

Mentor
Hello friends from afar.

I ran into what I felt to be somewhat of an odd question:

Prove that some odd numbers are primitive roots modulo pm for each odd prime p and each positive integer m.

It feels dodgy given that any odd number n = p1p2 ⋅⋅⋅ ps cannot be a primitive root of a prime number involved in its prime factorization. I just needed to be sure. Many thanks.
In future posts, please don't delete the homework template...