help please

I started this problem by noting that there has to be at least a 13.

1*2*3*4*5*6*7*8*9*10*11*12*13

This only has 2 5's so we go up to 15.

1*2*3*4*5*6*7*8*9*10*11*12*13*14*15

Not enough 2's so we use trial and error to find

1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16= 16!

Because there is a 7², we know that it has to be at least 14! and not as big as 21!

Starting at the bottom"

2! = 2¹

3! = 2¹3¹

4! = 2³3¹

5! = 2³3¹5¹

6! = 2⁴3²5¹

7! = 2⁴3²5¹7¹

8! = 2⁷3²5¹7¹ 

9! = 2⁷3⁴5¹7¹ 

10! = 2⁸3⁴5²7¹ 

11! = 2⁸3⁴5²7¹11¹ 

12! = 2¹⁰3⁵5²7¹11¹ 

13! = 2¹⁰3⁵5²7¹11¹13¹ 

14! = 2¹¹3⁵5²7²11¹13¹  

15! = 2¹¹3⁶5³7²11¹13¹  

16! = 2¹⁵3⁶5³7²11¹13¹