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Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?
- a)There will be no number left on the board.
- b)There will be exactly one number left on the board and it has a unique value.
- c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.
- d)There will be exactly two numbers left on the board.
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Natural numbers from 1 to 999 are written in a random manner on a boar...
There are total 1000 numbers. Each time 2 numbers are erased and replaced with their product, one number is getting reduced. After 999 such operations, 999 numbers would have got erased leaving behind exactly 1 number which means option ‘a’ or ‘b’ or ‘c’.
Most Upvoted Answer
Natural numbers from 1 to 999 are written in a random manner on a boar...
Analysis:
The given problem can be solved using the concept of prime factorization. Let's analyze each option one by one.
Option a) There will be no number left on the board:
This option is not possible because after each operation, the product of two numbers is written on the board. This process continues until only one number is left on the board.
Option b) There will be exactly one number left on the board and it has a unique value:
To understand this option, let's consider the prime factorization of the numbers from 1 to 999. The prime factorization of a number represents the product of its prime factors.
Prime Factorization:
- The prime factorization of a number can be represented as a product of prime numbers. For example, the prime factorization of 12 is 2 * 2 * 3.
Product of Two Numbers:
- When we choose two numbers and find their product, the prime factorization of the product will include all the prime factors of the chosen numbers.
- For example, if we choose 12 and 18, their product is 216. The prime factorization of 216 is 2 * 2 * 2 * 3 * 3 * 3.
Observation:
- The prime factorization of the product will always include all the prime factors of the chosen numbers.
- As the numbers from 1 to 999 have a finite set of prime factors, the prime factorization of the product after each operation will always include the same set of prime factors.
Conclusion:
- Based on the above observation, we can conclude that after 999 operations, only one number will be left on the board, and its value will be the product of all the prime factors raised to their respective powers.
Option c) There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen:
This option is not possible because the value of the number left on the board depends only on the prime factors of the chosen numbers, not on the order in which they are chosen.
Option d) There will be exactly two numbers left on the board:
This option is not possible because after each operation, the number of numbers on the board decreases by one. Since we started with 999 numbers, after 999 operations, there will be only one number left on the board.
Conclusion:
Based on the analysis, the correct answer is option 'b'. After 999 operations, there will be exactly one number left on the board, and its value will be unique.
The given problem can be solved using the concept of prime factorization. Let's analyze each option one by one.
Option a) There will be no number left on the board:
This option is not possible because after each operation, the product of two numbers is written on the board. This process continues until only one number is left on the board.
Option b) There will be exactly one number left on the board and it has a unique value:
To understand this option, let's consider the prime factorization of the numbers from 1 to 999. The prime factorization of a number represents the product of its prime factors.
Prime Factorization:
- The prime factorization of a number can be represented as a product of prime numbers. For example, the prime factorization of 12 is 2 * 2 * 3.
Product of Two Numbers:
- When we choose two numbers and find their product, the prime factorization of the product will include all the prime factors of the chosen numbers.
- For example, if we choose 12 and 18, their product is 216. The prime factorization of 216 is 2 * 2 * 2 * 3 * 3 * 3.
Observation:
- The prime factorization of the product will always include all the prime factors of the chosen numbers.
- As the numbers from 1 to 999 have a finite set of prime factors, the prime factorization of the product after each operation will always include the same set of prime factors.
Conclusion:
- Based on the above observation, we can conclude that after 999 operations, only one number will be left on the board, and its value will be the product of all the prime factors raised to their respective powers.
Option c) There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen:
This option is not possible because the value of the number left on the board depends only on the prime factors of the chosen numbers, not on the order in which they are chosen.
Option d) There will be exactly two numbers left on the board:
This option is not possible because after each operation, the number of numbers on the board decreases by one. Since we started with 999 numbers, after 999 operations, there will be only one number left on the board.
Conclusion:
Based on the analysis, the correct answer is option 'b'. After 999 operations, there will be exactly one number left on the board, and its value will be unique.
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Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer?
Question Description
Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer?.
Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer?.
Solutions for Natural numbers from 1 to 999 are written in a random manner on a board. Each time 2 numbers are chosen and their product is written on the board after erasing the 2 numbers. After 999 such operations, which of the following is likely to be true?a)There will be no number left on the board.b)There will be exactly one number left on the board and it has a unique value.c)There will be exactly one number left on the board whose value depends on the order in which the numbers are chosen.d)There will be exactly two numbers left on the board.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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