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The product of the predecessor and successor of an odd natural number is always divisible by
- a)2
- b)8
- c)6
- d)4
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The product of the predecessor and successor of an odd natural number ...
We know that the predecessor of an odd number is an even number and the successor of an odd number is also an even number.
So the two even numbers and their product are two consecutive even numbers which is always divisible by 8.
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The product of the predecessor and successor of an odd natural number ...
Understanding the Problem
To solve the problem, let's define an odd natural number. An odd number can be expressed in the form of 2n + 1, where n is a natural number. The predecessor (previous number) of this odd number is 2n, and the successor (next number) is 2n + 2.
Calculating the Product
Now, we calculate the product of the predecessor and the successor:
- Predecessor: 2n
- Successor: 2n + 2
The product is:
- 2n * (2n + 2) = 2n * 2(n + 1) = 4n(n + 1)
Analyzing the Divisibility
Now, let’s analyze the expression 4n(n + 1):
- 4: This factor shows that the product is always divisible by 4.
- n(n + 1): Since n and n + 1 are two consecutive integers, one of them is always even. Therefore, their product n(n + 1) is always even.
Putting it all together:
- The product 4n(n + 1) is divisible by both 4 and 2. However, since we are looking for a larger divisor, we focus on 8.
Conclusion
- The product of the predecessor and successor of an odd natural number is indeed divisible by 8 because:
- The factor 4 guarantees divisibility by 4.
- The even nature of n(n + 1) ensures that the product is divisible by an additional 2, confirming divisibility by 8.
Thus, the correct answer is option 'B', which indicates that the product is always divisible by 8.
To solve the problem, let's define an odd natural number. An odd number can be expressed in the form of 2n + 1, where n is a natural number. The predecessor (previous number) of this odd number is 2n, and the successor (next number) is 2n + 2.
Calculating the Product
Now, we calculate the product of the predecessor and the successor:
- Predecessor: 2n
- Successor: 2n + 2
The product is:
- 2n * (2n + 2) = 2n * 2(n + 1) = 4n(n + 1)
Analyzing the Divisibility
Now, let’s analyze the expression 4n(n + 1):
- 4: This factor shows that the product is always divisible by 4.
- n(n + 1): Since n and n + 1 are two consecutive integers, one of them is always even. Therefore, their product n(n + 1) is always even.
Putting it all together:
- The product 4n(n + 1) is divisible by both 4 and 2. However, since we are looking for a larger divisor, we focus on 8.
Conclusion
- The product of the predecessor and successor of an odd natural number is indeed divisible by 8 because:
- The factor 4 guarantees divisibility by 4.
- The even nature of n(n + 1) ensures that the product is divisible by an additional 2, confirming divisibility by 8.
Thus, the correct answer is option 'B', which indicates that the product is always divisible by 8.
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The product of the predecessor and successor of an odd natural number is always divisible bya)2b)8c)6d)4Correct answer is option 'B'. Can you explain this answer? for Class 6 2024 is part of Class 6 preparation. The Question and answers have been prepared according to the Class 6 exam syllabus. Information about The product of the predecessor and successor of an odd natural number is always divisible bya)2b)8c)6d)4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 6 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The product of the predecessor and successor of an odd natural number is always divisible bya)2b)8c)6d)4Correct answer is option 'B'. Can you explain this answer?.
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