Question

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

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Cristian

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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: Sincek^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 getk^3 = 3m - 1 .

Step 3: Now, observe thatk^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: Sincek+1 divides 3m - 1 and 3 divides k^3 + 1 , we can conclude that k+1 must divide k^3 + 1 as well.

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

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

Given:

We want to prove:

Step 1: Since

Step 2: Subtract 1 from both sides to get

Step 3: Now, observe that

Step 4: From Step 2,

Step 5: Since the right side of the equation is an integer,

Step 6: Since

Therefore, if

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