prove for k is an element of integers. 'If k^3+1 is divisible by 3 then k+1 is divisible by 3'

To prove the statement "If k^3 + 1 is divisible by 3, then k + 1 is divisible by 3" for k \in \mathbb{Z}, we can use direct proof.

Given: k^3 + 1 is divisible by 3.

We want to prove: k + 1 is divisible by 3.

Step 1: Since k^3 + 1 is divisible by 3, we can write k^3 + 1 = 3m for some integer m.

Step 2: Subtract 1 from both sides to get k^3 = 3m - 1.

Step 3: Now, observe that k^3 = (k+1)(k^2 - k + 1).

Step 4: From Step 2, 3m - 1 = (k+1)(k^2 - k + 1).

Step 5: Since the right side of the equation is an integer, k+1 divides 3m - 1.

Step 6: Since k+1 divides 3m - 1 and 3 divides k^3 + 1, we can conclude that k+1 must divide k^3 + 1 as well.

Therefore, if k^3 + 1 is divisible by 3, then k+1 is also divisible by 3.

\boxed{Answer}: k + 1 is divisible by 3.