Question

prove that the product of any three consecutive positive integers is divisible by 6 ​

Answer 1

Step-by-step explanation:

Let three consecutive positive integers be, n, n 1 and n+2.

When a number is divided by 3, the remainder obtained is

either 0 or 1 or 2

n 3p or 3p 1 or 3p + 2, where p is some integer

If n 3p, then n is divisible by 3.

If n

3p 1,n 2 3p +1+2 3p +3 3(p + 1 ) is

divisible by 3

If n 3p+2, n +1 3p + 2 +1 -3p + 3 = 3(p+1) is

divisible by 3

divisible by 3

So, we can say that one of the numbers among n, n + 1 and n

2 is always divisible by 3.

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

Similarly, when a number is divided 2, the remainder obtained

is 0 or 1

n-2q or 2q+1, where is some integer.

If n 2q n and n+2 2q+2 2(q+1) are divisible by 2

If n 2 q+1 n +1 2q + 1+1 2 q+ 2 2 (q + 1) is

divisible by 2