VIPSolutionsgive answer in 1 step with explanation at the end of each step and final answer at the end:QUESTION 6 The integer n?- nis even for all integers n. O True O False QUESTION 7 The integer n° - n is divisible by 3 for all integers n. O True O False QUESTION 8 The integer 6n is divisible by 4 for all integers n. O True O False
Question:
give answer in 1 step with explanation at the end of each step and final answer at the end:
QUESTION 6
The integer n?- nis even for all integers n.
O True
O False
QUESTION 7
The integer n° - n is divisible by 3 for all integers n.
O True
O False
QUESTION 8
The integer 6n is divisible by 4 for all integers n.
O True
O False
This is AI generated Response.
Asked by: Babu
Created at: 2025-07-11 07:14:43
Asked by: Babu
Created at: 2025-07-11 07:14:43
Final Answer:
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Let's answer each question in one step with explanations:
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**Question 6:**
The integer \( n^2 - n \) is even for all integers \( n \).
- **Step:** \( n^2 - n = n(n-1) \), which is the product of two consecutive integers. One of them is always even, so their product is always even.
- **Final Answer:** **True**
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**Question 7:**
The integer \( n^3 - n \) is divisible by 3 for all integers \( n \).
- **Step:** \( n^3 - n = n(n^2-1) = n(n-1)(n+1) \). Among any three consecutive integers, one is divisible by 3, so their product is always divisible by 3.
- **Final Answer:** **True**
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**Question 8:**
The integer \( 6n \) is divisible by 4 for all integers \( n \).
- **Step:** \( 6n \) is only divisible by 4 if \( n \) is even. For \( n = 1 \), \( 6 \) is not divisible by 4, so it's not true for all \( n \).
- **Final Answer:** **False**
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