The least common multiple of a pair of numbers is 8 means the numbers can be written as:

8x and 8y, where LCM(x, y) = 1, x, and y are integers

On the other hand the difference between the numbers is 6:

8x - 8y = 6

8(x - y) = 6 divide both sides by 2

4(x - y) = 3

x - y = 3/4

The difference between two integer numbers, x and y, is always an integer. Thus we have contradiction: x - y is integer and x - y = 3/4 is not integer, follows there is no such pair of numbers.