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Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain?
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Find the smallest number which can be expressed as the sum of two cube...
Option 'A' is right answer. which is 1729, also known as Hardy - Ramanujan number.
1729 =1728 + 1
1729 = 1000 +729
1729 =1728 + 1
1729 = 1000 +729
Community Answer
Find the smallest number which can be expressed as the sum of two cube...
Understanding the Problem
To find the smallest number that can be expressed as the sum of two cubes of natural numbers, we can represent it mathematically as:
\[ n = a^3 + b^3 \]
where \( a \) and \( b \) are natural numbers.
Calculating Cubes of Natural Numbers
Let's calculate the cubes of the first few natural numbers:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)
- \( 10^3 = 1000 \)
- \( 11^3 = 1331 \)
- \( 12^3 = 1728 \)
Finding the Sums of Two Cubes
We need to find combinations of these cubes that yield the smallest number:
- \( 1^3 + 12^3 = 1 + 1728 = 1729 \)
- \( 9^3 + 10^3 = 729 + 1000 = 1729 \)
Thus, \( 1729 \) can be expressed as the sum of two different pairs of cubes: \( 1^3 + 12^3 \) and \( 9^3 + 10^3 \).
Conclusion
After examining the given options:
- a) 1729
- b) 1001
- c) 1728
- d) 1332
The smallest number that can be expressed as the sum of two cubes of natural numbers is indeed option a) 1729. This number is famously known as the "Hardy-Ramanujan number," illustrating the beauty of number theory.
To find the smallest number that can be expressed as the sum of two cubes of natural numbers, we can represent it mathematically as:
\[ n = a^3 + b^3 \]
where \( a \) and \( b \) are natural numbers.
Calculating Cubes of Natural Numbers
Let's calculate the cubes of the first few natural numbers:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)
- \( 10^3 = 1000 \)
- \( 11^3 = 1331 \)
- \( 12^3 = 1728 \)
Finding the Sums of Two Cubes
We need to find combinations of these cubes that yield the smallest number:
- \( 1^3 + 12^3 = 1 + 1728 = 1729 \)
- \( 9^3 + 10^3 = 729 + 1000 = 1729 \)
Thus, \( 1729 \) can be expressed as the sum of two different pairs of cubes: \( 1^3 + 12^3 \) and \( 9^3 + 10^3 \).
Conclusion
After examining the given options:
- a) 1729
- b) 1001
- c) 1728
- d) 1332
The smallest number that can be expressed as the sum of two cubes of natural numbers is indeed option a) 1729. This number is famously known as the "Hardy-Ramanujan number," illustrating the beauty of number theory.
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Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain?
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Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain?.
Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the smallest number which can be expressed as the sum of two cubes of natural numbers Options a) 1729 b) 1001 c) 1728 d) 1332 correct answer is option a) how, explain?.
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