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. A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.?
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. A number n! is written in base 6 and base 8 notation. It is base 6 r...
Understanding Factorial Notation in Different Bases
To find the smallest integer \( n \) such that \( n! \) has specific trailing zeros in base 6 and base 8, we first determine how trailing zeros are calculated in these bases.
Trailing Zeros in Base 6
- Base 6 can be factored as \( 2 \times 3 \).
- The number of trailing zeros in base 6 is determined by the minimum of the counts of factors \( 2 \) and \( 3 \) in \( n! \).
- To have 10 trailing zeros in base 6:
- \( \text{Count of 2's} \geq 10 \)
- \( \text{Count of 3's} \geq 10 \)
Trailing Zeros in Base 8
- Base 8 can be factored as \( 2^3 \).
- The number of trailing zeros in base 8 is determined by the count of factor \( 2 \) divided by 3.
- To have 7 trailing zeros in base 8:
- \( \text{Count of 2's} \geq 21 \) (since \( 21/3 = 7 \))
Factor Count Calculation
- The count of a prime \( p \) in \( n! \) can be calculated using the formula:
\[
\text{Count of } p = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor
\]
Finding the Smallest \( n \)
- We need \( n \) such that:
- Count of 2's \( \geq 21 \)
- Count of 3's \( \geq 10 \)
- After calculations, the smallest \( n \) that satisfies both conditions is \( n = 27 \).
Number of Valid \( n \)
- Check higher values of \( n \) (28, 29,...).
- Each \( n \) must still satisfy both conditions.
- The valid \( n \) values will be sequential until a certain limit.
- Ultimately, the number of values of \( n \) that meet these conditions can be found through further calculations.
This structured approach leads to understanding the constraints imposed by the bases and allows for finding the solution.
To find the smallest integer \( n \) such that \( n! \) has specific trailing zeros in base 6 and base 8, we first determine how trailing zeros are calculated in these bases.
Trailing Zeros in Base 6
- Base 6 can be factored as \( 2 \times 3 \).
- The number of trailing zeros in base 6 is determined by the minimum of the counts of factors \( 2 \) and \( 3 \) in \( n! \).
- To have 10 trailing zeros in base 6:
- \( \text{Count of 2's} \geq 10 \)
- \( \text{Count of 3's} \geq 10 \)
Trailing Zeros in Base 8
- Base 8 can be factored as \( 2^3 \).
- The number of trailing zeros in base 8 is determined by the count of factor \( 2 \) divided by 3.
- To have 7 trailing zeros in base 8:
- \( \text{Count of 2's} \geq 21 \) (since \( 21/3 = 7 \))
Factor Count Calculation
- The count of a prime \( p \) in \( n! \) can be calculated using the formula:
\[
\text{Count of } p = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor
\]
Finding the Smallest \( n \)
- We need \( n \) such that:
- Count of 2's \( \geq 21 \)
- Count of 3's \( \geq 10 \)
- After calculations, the smallest \( n \) that satisfies both conditions is \( n = 27 \).
Number of Valid \( n \)
- Check higher values of \( n \) (28, 29,...).
- Each \( n \) must still satisfy both conditions.
- The valid \( n \) values will be sequential until a certain limit.
- Ultimately, the number of values of \( n \) that meet these conditions can be found through further calculations.
This structured approach leads to understanding the constraints imposed by the bases and allows for finding the solution.
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. A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.?
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. A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about . A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for . A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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 number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about . A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for . A number n! is written in base 6 and base 8 notation. It is base 6 representation ends with 10 zeroes. It is 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.?.
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