The product of any three consecutive natural numbers is always divisible by ______

Given:

P(n) = n(n + 1)(n + 2) is the product of three consecutive natural numbers.

Calculation:

Let n, n + 1, n + 2 be three consecutive natural numbers and P(n) be their product. Then,

P(n) = n(n + 1)(n + 2)

We have,

$P\left(1\right)=1\times 2\times 3=6$, which is divisible by 3 and 6.

$P\left(2\right)=2\times 3\times 4=24$, which is divisible by 3, 8 and 6.

$P\left(3\right)=3\times 4\times 5=60$, which is divisible by 3 and 6.

$P\left(4\right)=4\times 5\times 6=120$, which is divisible by 3, 8 and 6.

 

⇒ P(n) is divisible by 3 and 6 both.

 

∴ The product of any three consecutive natural numbers is always divisible by 6 and 3 both.

 

Alternate Method 

Calculation: 

 

The product of any three natural numbers is divisible by 6 or 3 both.

Let the three consecutive natural numbers be 1,2 and 3.

Their product is 6, which is divisible by 6

Let the other set of three consecutive natural numbers be 3, 4 and 5.

Their product is 60, which is divisible by 6 and 3.

∴ The product of any three consecutive natural numbers is always divisible by 6 and 3 both.