Question

Prove that the product of any three consecutive positive integers is divisible by 6. Solution: Let three consecutive numbers are n, n + 1, n + 2

Answer 1

Solution⤵

➡1st Case:-

If n is even

This means n + 2 is also even.

Hence n and n + 2 are divisible by 2

Also, product of n and (n + 2) is divisible by 2.

.’. n(n + 2) is divisible by 2.

This conclude n(n + 2) (n + 1) is divisible by 2 …(i)

As, n, n + 1, n + 2 are three consecutive numbers. n(n + 1) (n + 2) is a multiple of 3.

This shows n(n + 1) (n + 2) is divisible by 3. …(ii)

By equating (i) and (ii) we can say

n(n + 1) (n + 2) is divisible by 2 and 3 both.

Hence, n(n + 1) (n + 2) is divisible by 6.

➡2nd Case:-

When n is odd.

This show (n + 1) is even

Hence (n + 1) is divisible by 2. …(iii)

This conclude n(n + 1) (n + 2) is an even number and divisible by 2.

Also product of three consecutive number is a multiple of 3.

n(n + 1)(n + 2) is divisible by 3. …(iv)

Equating (iii) and (iv) we can say

n(n + 1) (n + 2) is divisible by both 2 and 3 Hence, n(n + 1)(n + 2) is divisible by 6.

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