Smallest Integer

7!   ˂ 10⁷                    8!      ˂    10⁸ 

Which is the smallest integer S so that S! > 10ˢ

Answer:  23!          25,852, 000 000 000 000 000 000

                  10²³       100 000 000 000 000 000 000 000

                 24!         620,450, 000 000 000 000 000 000

                 10²⁴       1,000 000 000 000 000 000 000 000

                  25!        15,511, 000 000 000 000 000 000 000

                 10²⁵         10,000 000 000 000 000 000 000 000

Thus the smallest S is 25.