Solution: Among any three consecutive integers, there is at least one multiple of 2 and one multiple of 3. Thus, the product is divisible by $ 2 \times 3 = 6 $. For example, $ 1 \times 2 \times 3 = 6 $, $ 2 \times 3 \times 4 = 24 $, and $ 3 \times 4 \times 5 = 60 $, all divisible by 6. No higher integer (e.g., 12) divides all such products, as $ 1 \times 2 \times 3 = 6 $ is not divisible by 12. - Abu Waleed Tea