Suppose x is the smallest integer greater than 3 such that when it is divided by 6 or 8, the remainder is 3. y is the smallest integer greater than 2 such that when it is divided by 6 or 8, the remainder is 2. What is x - y ?

To find x, we identify the smallest integer greater than 3 that leaves a remainder of 3 when divided by 6 and 8. This implies that (x - 3) must be a multiple of both 6 and 8. The Least Common Multiple (LCM) of 6 and 8 is 24. Thus, x can be expressed as 24k + 3. For the smallest x > 3, we set k = 1, which gives x = 24(1) + 3 = 27.

To find y, we identify the smallest integer greater than 2 that leaves a remainder of 2 when divided by 6 and 8. This implies that (y - 2) must be a multiple of the LCM of 6 and 8, which is 24. Thus, y = 24k + 2. For the smallest y > 2, we set k = 1, which gives y = 24(1) + 2 = 26.

The value of x - y = 27 - 26 = 1. Therefore, the correct option is A.