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The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M?
Most Upvoted Answer
The number 500! is successively divided by 35. P times, and then by 15...
Analysis:
To find the index of the highest power of 5 that divides M, we need to analyze the prime factorization of the number obtained after all the divisions.
Prime Factorization of 500!:
The prime factorization of 500! can be determined by counting the powers of 5 in the factorial. Since 5 is the smallest prime factor, we only need to count the powers of 5. The formula for calculating the power of a prime in n! is given by n/p + n/p^2 + n/p^3 + ..., where p is the prime number.
Divisions:
1. 500! is divided by 35 P times. This removes all the factors of 5 and 7 from the factorial.
2. Then it is divided by 15 Q times. This removes all the remaining factors of 3 and 5.
3. Finally, it is divided by 21 R times. This removes all the factors of 3 and 7.
Final Quotient M:
The final quotient M obtained after all the divisions will have the prime factorization with only the powers of 2 and 3. Since M is divisible by 3210 but not by 3211 and not by 7, it must have a power of 2 and 3 that cancels out the factors of 7 and 11.
Highest Power of 5 in M:
Since M does not have any factors of 5, the index of the highest power of 5 that divides M will be 0.
Therefore, the index of the highest power of 5 that divides M is 0.
To find the index of the highest power of 5 that divides M, we need to analyze the prime factorization of the number obtained after all the divisions.
Prime Factorization of 500!:
The prime factorization of 500! can be determined by counting the powers of 5 in the factorial. Since 5 is the smallest prime factor, we only need to count the powers of 5. The formula for calculating the power of a prime in n! is given by n/p + n/p^2 + n/p^3 + ..., where p is the prime number.
Divisions:
1. 500! is divided by 35 P times. This removes all the factors of 5 and 7 from the factorial.
2. Then it is divided by 15 Q times. This removes all the remaining factors of 3 and 5.
3. Finally, it is divided by 21 R times. This removes all the factors of 3 and 7.
Final Quotient M:
The final quotient M obtained after all the divisions will have the prime factorization with only the powers of 2 and 3. Since M is divisible by 3210 but not by 3211 and not by 7, it must have a power of 2 and 3 that cancels out the factors of 7 and 11.
Highest Power of 5 in M:
Since M does not have any factors of 5, the index of the highest power of 5 that divides M will be 0.
Therefore, the index of the highest power of 5 that divides M is 0.
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The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M?
Question Description
The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M?.
The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M?.
Solutions for The number 500! is successively divided by 35. P times, and then by 15, Q times and finally by 21, R times where P, Q, R are positive integers. These divisions have no remainders at any stage. If the final quotient (M) thus obtained is divisible by 3210 but not by 3211, and also not divisible by 7, which of the following numbers could be the index of the highest power of 5 that divides M? in English & in Hindi are available as part of our courses for CAT.
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