an army of men can line up to march in rows of 1,2,3,4,5,6,7,8,9 or 10 and there is never any rows that is incomplete. what is the smallest number of soldiers in the army?

I was always taught to multiply the highest powers of each prime factor. Using Arthur's list they would be

2

^{3}3

^{2}5

^{1}7

^{1}which would give you 8x9x5x7 = 40x63 = 4x630 = 2x1260 = 2520