Question

Prove that the product of three consecutive positive integer is divisible by six

Answer 1

heya...!!!!.

let the no. be (x) , (x + 1) ,(x + 2).

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

therefore,

x = 3n or (3n + 1) or (3n + 2)

if x = 3n, then x is divisible by 3

if x = 3n + 1 ,then x + 2 = 3n + 1 + 2 = 3n + 3

=> x = 3(n + 1) is divisible by 3

if x = 3n + 2, then x + 1 = 3n + 2 + 1

=> 3n + 3 = 3(n + 1)

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

 n (n + 1) (n + 2) is divisible by 3.

now,

similarly, when a no. is divisible by 2 remainder abtained is 0 or 1.

therefore,

x = 2r or (2r + 1)

if x = 2r, then x = 2r and (x + 2) then,2r + 2

=> 2(r + 1) are divisible by 2

if x = (2r + 1), then x + 1 = 2r + 1 + 1 = 2(r + 1)is divisible by 2.

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

 x (x + 1) (x + 2) is divisible by 2.

 

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

 

therefore,

 n (n + 1) (n + 2) is divisible by 6.