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The cube of an odd number is always __________.
- a)odd number
- b)even number
- c)prime number
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The cube of an odd number is always __________.a)odd number b)even ...
The cube of a natural odd number is always odd.
For natural numbers, we know,
odd x odd = odd
odd x even = even
even x even = even.
So, cube of a natural odd number is
odd x odd x odd
= odd x odd
= odd.
Eg.: Cube of an odd number 3 is 27, which is also odd.
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For natural numbers, we know,
odd x odd = odd
odd x even = even
even x even = even.
So, cube of a natural odd number is
odd x odd x odd
= odd x odd
= odd.
Eg.: Cube of an odd number 3 is 27, which is also odd.
Most Upvoted Answer
The cube of an odd number is always __________.a)odd number b)even ...
The Cube of an Odd Number
The statement that the cube of an odd number is always odd can be understood through the properties of odd and even numbers. Let's delve into the details.
Understanding Odd and Even Numbers
- **Definition of Odd Numbers**: An odd number can be expressed in the form \(2n + 1\), where \(n\) is an integer. Examples include -1, 1, 3, 5, 7, etc.
- **Definition of Even Numbers**: An even number can be expressed as \(2n\), where \(n\) is an integer. Examples include -2, 0, 2, 4, 6, etc.
Calculating the Cube
When an odd number is cubed, the calculation can be represented as follows:
- Let \(x = 2n + 1\) (an odd number).
- The cube of \(x\) is calculated as:
\[
x^3 = (2n + 1)^3
\]
Using the binomial theorem, we expand this:
\[
(2n + 1)^3 = (2n)^3 + 3(2n)^2(1) + 3(2n)(1^2) + 1^3
\]
This simplifies to:
\[
= 8n^3 + 12n^2 + 6n + 1
\]
- **Key Observation**: The last term, \(1\), confirms that the entire expression is odd, as any sum of even numbers (like \(8n^3\), \(12n^2\), and \(6n\)) remains even, and adding \(1\) results in an odd number.
Conclusion
- Therefore, the cube of any odd number \(x\) results in another odd number. This confirms that the correct answer to the statement is:
- **a) odd number**.
This mathematical property holds for all odd integers, making it a reliable characterization.
The statement that the cube of an odd number is always odd can be understood through the properties of odd and even numbers. Let's delve into the details.
Understanding Odd and Even Numbers
- **Definition of Odd Numbers**: An odd number can be expressed in the form \(2n + 1\), where \(n\) is an integer. Examples include -1, 1, 3, 5, 7, etc.
- **Definition of Even Numbers**: An even number can be expressed as \(2n\), where \(n\) is an integer. Examples include -2, 0, 2, 4, 6, etc.
Calculating the Cube
When an odd number is cubed, the calculation can be represented as follows:
- Let \(x = 2n + 1\) (an odd number).
- The cube of \(x\) is calculated as:
\[
x^3 = (2n + 1)^3
\]
Using the binomial theorem, we expand this:
\[
(2n + 1)^3 = (2n)^3 + 3(2n)^2(1) + 3(2n)(1^2) + 1^3
\]
This simplifies to:
\[
= 8n^3 + 12n^2 + 6n + 1
\]
- **Key Observation**: The last term, \(1\), confirms that the entire expression is odd, as any sum of even numbers (like \(8n^3\), \(12n^2\), and \(6n\)) remains even, and adding \(1\) results in an odd number.
Conclusion
- Therefore, the cube of any odd number \(x\) results in another odd number. This confirms that the correct answer to the statement is:
- **a) odd number**.
This mathematical property holds for all odd integers, making it a reliable characterization.
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The cube of an odd number is always __________.a)odd number b)even ...
The cube of an odd number is always an odd number because:
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The cube of an odd number is always __________.a)odd number b)even numberc)prime number d)none of theseCorrect answer is option 'A'. Can you explain this answer?
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