If G has an element of order p and an element of order q whe

If G has an element of order p and an element of order q, where p and q are distinct primes, then the order of G is a multiple of pq.

Solution

More generally, if G has an element of order p and an element of order q, then by Lagrange\'s theorem the order of G is a multiple of both p and q and so is a multiple of lcm(p,q). In particular, if p and q are coprime, then ,lcm(p,q)=pq, and the order of G is a multiple of pq

If p and q are distinct primes, then they are coprime and the result above holds.