For N=5, Herman can choose a number from {1, 2, 3}. Observe how the scenario will play out for each choice:
1) one coprime pair (5, 1)
Hannah: 5
Herman: 1
2) two coprime pairs (5, 2), (4, 1)
Hannah: 5->4
Herman: 2->1
3) two coprime pairs (5, 3), (3, 1)
Hannah: 5->4->3
Herman: 3->2->1

Both 2 and 3 are co-optimal, so since 3 is the largest we choose that one.