Question

prove that n⁵+n³+n is divisible by 3​

Answer 1

Answer:

n

3

−n=n(n

2

−1)=n(n−1)(n+1)

Whenever 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, then n–1=3p+1–1=3p is divisible by 3.

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

So, we can say that one of the numbers among n,n–1and n+1 is always divisible by 3.

⇒n(n–1)(n+1) is divisible by 3.