Show that in any group of order 100 , either every element has order that is a power of a prime or there is an element of order 10. Without using Sylow's theorem how to approach this problem ? By Lagrange's theorem , possible orders of elements of G are 1,2,4,5,10,20,25,50 and 100. Any hint or suggestion will be appreciated.
Hint: You're on the right track. The list contains only $10, 20, 50, 100$ which are not prime powers. If e.g. $x$ has order $50$, then $x^5$ has order $10$.