Question

prove that the product of three consecutive positive integers is divisible by 6​

Answer 1

Hope this solution will help you

Answer 2

To prove:-

• The product of three consecutive positive integers is divisible by 6

Proof:-

• Let n be any positive integer.

Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5.

If n = 6q

=> n(n + 1)(n + 2) = 6q(6q + 1)(6q + 2), which is divisible by 6.

If n = 6q + 1

=> n(n + 1)(n + 2) = (6q + 1)(6q + 2)(6q + 3)

=> n(n + 1)(n + 2) = 6(6q + 1)(3q + 1)(2q + 1)

Which is divisible by 6.

If n = 6q + 2

=> n(n + 1)(n + 2) = (6q + 2)(6q + 3)(6q + 4)

=> n(n + 1)(n + 2) = 12(3q + 1)(2q + 1)(2q + 3)

Which is divisible by 6.

Similarly we can prove others.

Hence, it is proved that the product of three consecutive positive integers is divisible by 6.