Prove that product if three consecutive positive integers is divisible

1.	Prove that product if three consecutive positive integers is divisible by 3.
Answer» Answer: Step-by-step explanation: Let N be any POSITIVE integer.Let n (n +1) and (n + 2) are three consecutive positive integers. Then their PRODUCT is n (n +1)(n+2). Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5. Case : 1 If n = 6q, n (n + 1) (n + 2) = 6q (6q + 1) (6q + 2), which is DIVISIBLE by 6 Case : 2 If n = 6q + 1 n (n + 1) (n + 2) = (6q + 1) (6q + 2) (6q + 3) n (n + 1) (n + 2) = (6q + 1) 2(3q + 2) 3(2q + 1) n (n + 1) (n + 2) = 6 (6q + 1) (3q + 1) (2q + 1) ,Which is divisible by 6 Case : 3 If n = 6q + 2 n (n + 1) (n + 2) = (6q + 2) (6q + 3) (6q + 4) n (n + 1) (n + 2) = 2(3q + 1) 3(2q + 1) 2(3q + 2) n (n + 1) (n + 2) = 12 (3q + 1) (2q + 1) (3q + 2), Which is divisible by 6. Similarly, n (n + 1) (n + 2) is divisible by 6 if n= 6q + 3 or 6q + 4, 6q + 5. Hence it is proved that the product of three consecutive positive integers is divisible by 6.

Prove that product if three consecutive positive integers is divisible by 3.

Answer»

Answer:

Step-by-step explanation:

Let N be any POSITIVE integer.Let n (n +1) and (n + 2) are three consecutive positive integers. Then their PRODUCT is n (n +1)(n+2).

Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5.

Case : 1

If n = 6q,

n (n + 1) (n + 2) = 6q (6q + 1) (6q + 2), which is DIVISIBLE by 6

Case : 2

If n = 6q + 1

n (n + 1) (n + 2) = (6q + 1) (6q + 2) (6q + 3)

n (n + 1) (n + 2) = (6q + 1) 2(3q + 2) 3(2q + 1)

n (n + 1) (n + 2) = 6 (6q + 1) (3q + 1) (2q + 1) ,Which is divisible by 6

Case : 3

If n = 6q + 2

n (n + 1) (n + 2) = (6q + 2) (6q + 3) (6q + 4)

n (n + 1) (n + 2) = 2(3q + 1) 3(2q + 1) 2(3q + 2)

n (n + 1) (n + 2) = 12 (3q + 1) (2q + 1) (3q + 2), Which is divisible by 6.

Similarly, n (n + 1) (n + 2) is divisible by 6 if n= 6q + 3 or 6q + 4, 6q + 5.

Hence it is proved that the product of three consecutive positive integers is divisible by 6.