Question

show that product of three consecutive natural numbers is divisible by 6

Answer 1

Let number be x
So expression is x^3/6

Answer 2

Solution:

Let three consecutive positive integers be, n, n + 1 and n + 2. 

When a number is divided by 3, the remainder obtained is either 0 or 1 or 2.  

∴ n = 3p or 3p + 1 or 3p + 2, where p is some integer.

If n = 3p, then n is divisible by 3. 

If n = 3p + 1, ⇒ n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3. 

If n = 3p + 2, ⇒ n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3. 

So, we can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 3.  

⇒ n (n + 1) (n + 2) is divisible by 3. 

Similarly, when a number is divided 2, the remainder obtained is 0 or 1. 

∴ n = 2q or 2q + 1, where q is some integer. 

If n = 2q ⇒ n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2. 

If n = 2q + 1 ⇒ n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2. 

So, we can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 2. 

⇒ n (n + 1) (n + 2) is divisible by 2. 

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

∴ n (n + 1) (n + 2) is divisible by 6.