Question

The product of 3 consecutive positive integers is divisible by 6 prove. Tell statement is true or false

Answer 1

Answer:True

Step-by-step explanation:

Let us consider three consecutive  integers be, x, x + 1 and x + 2.

When we divide a number by 3 the remainder  is either 0 or 1 or 2.

let x = 3p or 3p + 1 or 3p + 2, where p is some integer.

If x = 3p, then x is divisible by 3.

If x = 3p + 1, then x + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.

If x = 3p + 2, then x + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.

So that x, x + 1 and x + 2 is always divisible by 3.

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

Similarly, When we divide a number by 2 the remainder  is either 0 or 1.

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

If x = 2q, then x and x + 2 = 2q + 2 = 2(q + 1) are divisible by 2.

If x = 2q + 1, then x + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.

So that x, x + 1 and x + 2 is always divisible by 2.

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

But x (x + 1) (x + 2) is divisible by 2 and 3.

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