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31 Jul 2025, 08:26
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If a positive integer \(n\) has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that \(n\) can have?
A. 1
B. 2
C. 3
D. 4
E. 5
A. 1
B. 2
C. 3
D. 4
E. 5
31 Jul 2025, 08:26
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Official Solution:
If a positive integer \(n\) has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that \(n\) can have?
A. 1
B. 2
C. 3
D. 4
E. 5
If the prime factorization of a positive integer is \(a^x * b^y * c^z *...\) where \(a\), \(b\), and \(c\) are prime numbers and \(x\), \(y\), and \(z\) are their powers, then the number of positive factors of the integer is given by \((x + 1)(y + 1)(z + 1)...\) So, to find the number of positive factors, we add 1 to the powers of the distinct primes in the prime factorization and multiply.
We are given that \(n\) has 24 factors. To find how many prime factors \(n\) can have, we consider how 24 can be written as a product of such terms.
The least number of primes \(n\) can have is 1, if \(n = prime^{23}\), which gives the number of factors as 23 + 1 = 24. For example, \(n\) could be \(2^{23}\).
For the maximum number of primes, break 24 into the maximum number of integers greater than 1: \(24 = 2 * 2 * 2 * 3\). In this case, \(n\) would have 4 primes: \(n = (prime_1) * (prime_2) * (prime_3) * (prime_4)^2\), which gives the number of factors as \((1 + 1)(1 + 1)(1 + 1)(2 + 1) = 2 * 2 * 2 * 3 = 24\). For example, \(n\) could be \(2 * 3 * 5 * 7^2\).
So, the range is 4 - 1 = 3.
Answer: C
If a positive integer \(n\) has 24 positive factors, what is the maximum possible range in the number of distinct prime factors that \(n\) can have?
A. 1
B. 2
C. 3
D. 4
E. 5
If the prime factorization of a positive integer is \(a^x * b^y * c^z *...\) where \(a\), \(b\), and \(c\) are prime numbers and \(x\), \(y\), and \(z\) are their powers, then the number of positive factors of the integer is given by \((x + 1)(y + 1)(z + 1)...\) So, to find the number of positive factors, we add 1 to the powers of the distinct primes in the prime factorization and multiply.
We are given that \(n\) has 24 factors. To find how many prime factors \(n\) can have, we consider how 24 can be written as a product of such terms.
The least number of primes \(n\) can have is 1, if \(n = prime^{23}\), which gives the number of factors as 23 + 1 = 24. For example, \(n\) could be \(2^{23}\).
For the maximum number of primes, break 24 into the maximum number of integers greater than 1: \(24 = 2 * 2 * 2 * 3\). In this case, \(n\) would have 4 primes: \(n = (prime_1) * (prime_2) * (prime_3) * (prime_4)^2\), which gives the number of factors as \((1 + 1)(1 + 1)(1 + 1)(2 + 1) = 2 * 2 * 2 * 3 = 24\). For example, \(n\) could be \(2 * 3 * 5 * 7^2\).
So, the range is 4 - 1 = 3.
Answer: C
15 Nov 2025, 23:29
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I like the solution - it’s helpful. Wow Just read that the question states range in the Maximum possible and hence I got my answer wrong! Great Question.
