Exercise
Suppose p and r are distinct primes dividing the order of the group G. Does G have an element of order pr?
yes
no, not in general, although there are groups G for which the statement
is true
no, never.
Wrong. The group D10 is a counterexample: it has no element of order 10.
(If it did, it would be isomorphic to the cyclic group of order 10.)
Correct.
No, think of a cyclic group of order pr