In this problem, we prove an important property of consecutive positive integers. We are required to show that the product of any three consecutive positive integers is always divisible by 6.
Question
Prove that the product of three consecutive positive integers is divisible by 6.
Solution
Let the three consecutive positive integers be
n, n + 1, and n + 2, where n is a positive integer.
The product of these three consecutive integers is
n × (n + 1) × (n + 2).
Among any three consecutive integers, one integer is always divisible by 3.
Also, among any two consecutive integers, one must be even. Therefore, among three consecutive integers, at least one integer is divisible by 2.
Hence, the product
n × (n + 1) × (n + 2)
is divisible by both 2 and 3.
Since 6 = 2 × 3, the product of three consecutive positive integers is divisible by 6.
Conclusion
Therefore, the product of three consecutive positive integers is always divisible by 6.
Hence proved.