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A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.
- a)24 and 3
- b)22 and 4
- c)25 and 4
- d)25 and 3
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A number n! is written in base 6 and base 8 notation. Its base 6 repr...
To find the smallest value of n that satisfies the given conditions, we need to analyze the base 6 and base 8 representations separately.
Base 6 Representation:
The base 6 representation of n! ends with 10 zeroes. This means that n! is divisible by 6^10. To find the smallest n that satisfies this condition, we can find the smallest power of 6 that divides n!.
The prime factorization of 6 is 2 * 3. So, to find the power of 6 in n!, we need to find the power of 2 and 3 separately.
Power of 2 in n!:
Using Legendre's formula, the power of 2 in n! is given by the sum of the quotients when n is divided by successive powers of 2.
The largest power of 2 that divides n is given by the highest power of 2 that is less than or equal to n. In this case, it is 2^5 = 32.
The sum of the quotients when n is divided by the powers of 2 is:
n/2 + n/4 + n/8 + n/16 + n/32.
Power of 3 in n!:
Similarly, the power of 3 in n! is given by the sum of the quotients when n is divided by successive powers of 3.
The largest power of 3 that divides n is given by the highest power of 3 that is less than or equal to n. In this case, it is 3^3 = 27.
The sum of the quotients when n is divided by the powers of 3 is:
n/3 + n/9 + n/27.
To satisfy the condition that n! is divisible by 6^10, the power of 2 and 3 in n! should be at least 10.
By trial and error, we can find that n = 24 satisfies this condition. The power of 2 in 24! is 22, and the power of 3 is 10.
Base 8 Representation:
The base 8 representation of n! ends with 7 zeroes. This means that n! is divisible by 8^7. To find the smallest n that satisfies this condition, we can find the smallest power of 8 that divides n!.
The prime factorization of 8 is 2^3. So, to find the power of 8 in n!, we need to find the power of 2 separately.
Power of 2 in n!:
Using Legendre's formula, the power of 2 in n! is given by the sum of the quotients when n is divided by successive powers of 2.
The largest power of 2 that divides n is given by the highest power of 2 that is less than or equal to n. In this case, it is 2^6 = 64.
The sum of the quotients when n is divided by the powers of 2 is:
n/2 + n/4 + n/8 + n/16 + n/32 + n/64.
To satisfy the condition that n! is divisible by 8^7, the power of 2 in n! should be at least 7.
By trial and error, we can find that n = 24 satisfies this condition. The power of 2 in 24! is
Base 6 Representation:
The base 6 representation of n! ends with 10 zeroes. This means that n! is divisible by 6^10. To find the smallest n that satisfies this condition, we can find the smallest power of 6 that divides n!.
The prime factorization of 6 is 2 * 3. So, to find the power of 6 in n!, we need to find the power of 2 and 3 separately.
Power of 2 in n!:
Using Legendre's formula, the power of 2 in n! is given by the sum of the quotients when n is divided by successive powers of 2.
The largest power of 2 that divides n is given by the highest power of 2 that is less than or equal to n. In this case, it is 2^5 = 32.
The sum of the quotients when n is divided by the powers of 2 is:
n/2 + n/4 + n/8 + n/16 + n/32.
Power of 3 in n!:
Similarly, the power of 3 in n! is given by the sum of the quotients when n is divided by successive powers of 3.
The largest power of 3 that divides n is given by the highest power of 3 that is less than or equal to n. In this case, it is 3^3 = 27.
The sum of the quotients when n is divided by the powers of 3 is:
n/3 + n/9 + n/27.
To satisfy the condition that n! is divisible by 6^10, the power of 2 and 3 in n! should be at least 10.
By trial and error, we can find that n = 24 satisfies this condition. The power of 2 in 24! is 22, and the power of 3 is 10.
Base 8 Representation:
The base 8 representation of n! ends with 7 zeroes. This means that n! is divisible by 8^7. To find the smallest n that satisfies this condition, we can find the smallest power of 8 that divides n!.
The prime factorization of 8 is 2^3. So, to find the power of 8 in n!, we need to find the power of 2 separately.
Power of 2 in n!:
Using Legendre's formula, the power of 2 in n! is given by the sum of the quotients when n is divided by successive powers of 2.
The largest power of 2 that divides n is given by the highest power of 2 that is less than or equal to n. In this case, it is 2^6 = 64.
The sum of the quotients when n is divided by the powers of 2 is:
n/2 + n/4 + n/8 + n/16 + n/32 + n/64.
To satisfy the condition that n! is divisible by 8^7, the power of 2 in n! should be at least 7.
By trial and error, we can find that n = 24 satisfies this condition. The power of 2 in 24! is
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Community Answer
A number n! is written in base 6 and base 8 notation. Its base 6 repr...
Base 6 representation ends with 10 zeroes, or the number is a multiple of 610. If n! has to be a multiple of 610, it has to be a multiple of 310. The smallest factorial that is a multiple of 310 is 24!. So, when n = 24, 25 or 26, n! will be a multiple of 610 (but not 611).
Similarly, for the second part, we need to find n! such that it is a multiple of 221, but not 224. When n = 24, n! is a multiple of 222. S0, when n = 24, 25, 26, 27, n! will be a multiple of 221 but not 224.
The smallest n that satisfies the above conditions is 24. n = 24, 25 or 26 will satisfy the above conditions.
The answer is "24 and 3".
Hence, the correct option is (a).
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A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer?
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A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer?.
A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.a)24 and 3b)22 and 4c)25 and 4d)25 and 3Correct answer is option 'A'. Can you explain this answer?.
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