give answer in 2 step with explanation at the end of each step and final answer at the end:QUESTION 7
The integer n° - n is divisible by 3 for all integers n.
O True
O False

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:QUESTION 7
The integer n° - n is divisible by 3 for all integers n.
O True
O False

VIPSolutions · Accepted Answer

**Step 1:**  
Let's factor the expression $ n^3 - n $:
$$
n^3 - n = n(n^2 - 1) = n(n-1)(n+1)
$$
This is the product of three consecutive integers.

**Explanation:**  
Any three consecutive integers always include a multiple of 3, so their product is always divisible by 3.

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**Step 2:**  
Since for any integer $ n $, $ n(n-1)(n+1) $ is always divisible by 3, it follows that $ n^3 - n $ is always divisible by 3.

**Explanation:**  
No matter what integer you pick, one of $ n, n-1, n+1 $ will be a multiple of 3, ensuring divisibility by 3.

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**Final answer:**  
**True**

✨ VIPSolutions

give answer in 2 step with explanation at the end of each step and final answer at the end:QUESTION 7 The integer n° - n is divisible by 3 for all integers n. O True O False

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