Question

(n^3_n) is divisible by 3. Explain the reason
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Answer 1

For n=1,

n3−n=1−1 which is divisible by 3

Assume the statement is true for some number n, that is, n3−n is divisible by 3. Now,

(n+1)3−(n+1)=n3+3n2+3n+1−n−1=(n3−n)+3(n2+n)=(n3−n)+3n(n+1)

which is n3−n plus a multiple of 3.

Since we assumed that n3−n was a multiple of 3, it follows that (n+1)3−(n+1) is also a multiple of 3.

So, since the statement "n3−n is divisible by 3" is true for n=1, and its truth for n implies its truth for n+1, the statement is true for all whole number n.

Answer 2

Answer:

Step-by-step explanation:

n ^3 - n = n(n^2 - 1) = n(n - 1)(n + 1).

Now

n - 1 , n , n+ 1 are Three consecutive positive integers, thus exactly one of them is divisible by 3 and at least one of them is divisible by 2. Since

hcf(2 , 3) = 1, then n(n - 1)(n + 1) is divisible by 6 and clearly then divisible by 3.