Sylow $p$-Subgroups and Counting Prime Numbers

Question

Use the Sylow Theorems to find the number of prime numbers $p$ and $q$ which satisfy the following conditions:

1. $p<q$

2. $q\not\equiv1($mod $p)$,
   

3. $pq<100$
   

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How would the Sylow Theorems apply to this problem? I'm really confused on how to use them in this setting. This problem just seems like a number theory problem that could be solved using Fermat's Theorem. I genuinely don't understand how to use Sylow $p$-subgroups to solve this problem. Any hints, or explanations would be extremely helpful since I find the Sylow Theorems hard to understand in general.

Answer 1

By Sylow's theorems, this is all such pairs such that all groups of order $pq$ are cyclic. I suppose you could then look at a list of all groups of each order to see which of these had only one. I agree it's a perversely stated problem.