Here's a hint (that I found helpful):
Consider an element of A_{10} as a permutation written in disjoint cycle notation. The lengths of the cycles must add up to no more than 10, since the permutations are of degree 10. Odd cycles have lengths 2, 4, 6, 8. Even cycles have lengths 3, 5, 7, 9. Since we're dealing with the alternating group, odd cycles must occur in pairs, otherwise you would have an odd permutation (not an even one). Determine the combination of cycle lengths that add up to no more than 10, form an even permutation, and have the largest LCM.
Enjoy!
