Prove that n^3+2 is not divisible by 8

Given

n3+ 2 is not divisible by 8 ∀ n ∈ N

Proof

For value of n, we have 2 cases

CASE 1 : n is odd

If n is odd then n3 is also odd. We know ODD + EVEN = ODD. Therefore, n3+2 is odd. 8 cannot divide an odd number. We can say that n3+ 2 is not divisible by 8 ∀ n ∈ O.

CASE 2 : n is even

If n is even, then we can write n as

n = 2px, p ∈ N

n3+ 2 = (2px)3+ 2

= 23p * x3 + 2

= 2 ( 23p-1 * x3 + 1 )

The term 23p-1 * x3 is even. Therefore, 23p-1 * x3 + 1 is odd. Thus, 2 ( 23p-1 * x3 + 1 ) cannot be divisible by 8. We can say that n3+ 2 is not divisible by 8 ∀ n ∈ E.

From the above two cases, we can conclude that n is not divisible by 8 ∀ n ∈ N.

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