What is the smallest possible counting number divisible by 1, 2, 3, 4, 5, 6, 7, and 8?

Question

0 is not a counting #

Answer 1

2 = 2
3 = 3
4 = 2*2
5 = 5
6 = 2*3
7 = 7
8 = 2*2*2

You need a sufficient number of each prime to cover all of your divisors.
You need three 2's (to cover 8)
You need one each of 3, 5 and 7.

Your answer = 2*2*2*3*5*7

Any number smaller than that (for example, by removing a 2) will miss being divisible by one or the other number in your list.

840/8 = 105
840/7 = 120
840/6 = 140
...

Answer 2

3 x 5 x 7 x 8 = 840