beame14 Prove that the product of three consecutive positive integer I

1.	beame14 Prove that the product of three consecutive positive integer Is dvisivi DJnoritive integer n, prove that n3 n divisible by 6brfis she wants to
Answer» Let us three consecutive integers be, x, x + 1 and x + 2 respectively. [• Whenever a number is divided by 3 the remainder obtained is either 0 or 1 or 2. ( it's proved )] Let x = 3p ........................(i)where p belongs to an integer and does not equal to zero ( 0 ). => x is divisible by 3. If x = 3p + 1 where p belongs to an integer and does not equal to zero ( 0 ). then x + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1)which is divisible by 3. If x = 3p + 2where p belongs to an integer and does not equal to zero ( 0 ). then x + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1)which 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, whenever a number is divided 2 we will get the remainder is 0 or 1. Let x = 2qwhere q belongs to an integer and does not equal to zero ( 0 ). => x is divisible by 2 If x = 2q + 1 where q belongs to an integer and does not equal to zero ( 0 ). then x + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) which 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. Since x (x + 1) (x + 2) is divisible by 2 and 3.∴ x (x + 1) (x + 2) is divisible by 6.

beame14 Prove that the product of three consecutive positive integer Is dvisivi DJnoritive integer n, prove that n3 n divisible by 6brfis she wants to

Answer»

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

[• Whenever a number is divided by 3 the remainder obtained is either 0 or 1 or 2. ( it's proved )]

Let x = 3p ........................(i)where p belongs to an integer and does not equal to zero ( 0 ).

=> x is divisible by 3.

If x = 3p + 1 where p belongs to an integer and does not equal to zero ( 0 ).

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

If x = 3p + 2where p belongs to an integer and does not equal to zero ( 0 ).

then x + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1)which 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, whenever a number is divided 2 we will get the remainder is 0 or 1.

Let x = 2qwhere q belongs to an integer and does not equal to zero ( 0 ).

=> x is divisible by 2

If x = 2q + 1 where q belongs to an integer and does not equal to zero ( 0 ).

then x + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) which 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.

Since x (x + 1) (x + 2) is divisible by 2 and 3.∴ x (x + 1) (x + 2) is divisible by 6.