Let the number be N

 1 = 1     so N must have factor 1 (every integer has factor 1, so we can ignore 1)
 2 = 2     so N must have factor 2
 3 = 3     so N must have factor 3
 4 = 2*2   so N must have factor 2*2  
 5 = 5     so N must have factor 5
 6 = 2*3   so N must have factor 2*3
 7 = 7     so N must have factor 7
 8 = 2*2*2 so N must have factor 2*2*2
 9 = 3*3   so N must have factor 3*3
10 = 2*5   so N must have factor 2*5

So the smallest N can be and have at least as many of 
every factor above is the product of three 2's, two 3's, 
one 5, and one 7, which means

N = 2*2*2*3*3*5*7 = 2520

Edwin