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16 Sep 2014, 00:54
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If \(x\) is a prime number, how many distinct positive factors does \(x^3\) have?
A. 2
B. 3
C. 4
D. 5
E. 6
A. 2
B. 3
C. 4
D. 5
E. 6
16 Sep 2014, 00:54
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Official Solution:
If \(x\) is a prime number, how many distinct positive factors does \(x^3\) have?
A. 2
B. 3
C. 4
D. 5
E. 6
Since \(x\) is a prime number, \(x^3\) will have 4 distinct positive divisors: 1, \(x\), \(x^2\), and \(x^3\). For example, \(2^3\) has the factors 1, 2, 4, and 8; similarly, \(3^3\) has the factors 1, 3, 9, and 27.
Finding the Number of Factors of an Integer
First, make the prime factorization of an integer \(n = a^p * b^q * c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\), and \(p\), \(q\), and \(r\) are their respective powers.
The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and \(n\) itself.
Example: Finding the number of all factors of 450: \(450 = 2^1 * 3^2 * 5^2\)
The total number of factors of 450, including 1 and 450 itself, is \((1+1)(2+1)(2+1) = 2*3*3 = 18\) factors.
Answer: C
If \(x\) is a prime number, how many distinct positive factors does \(x^3\) have?
A. 2
B. 3
C. 4
D. 5
E. 6
Since \(x\) is a prime number, \(x^3\) will have 4 distinct positive divisors: 1, \(x\), \(x^2\), and \(x^3\). For example, \(2^3\) has the factors 1, 2, 4, and 8; similarly, \(3^3\) has the factors 1, 3, 9, and 27.
Finding the Number of Factors of an Integer
First, make the prime factorization of an integer \(n = a^p * b^q * c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\), and \(p\), \(q\), and \(r\) are their respective powers.
The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and \(n\) itself.
Example: Finding the number of all factors of 450: \(450 = 2^1 * 3^2 * 5^2\)
The total number of factors of 450, including 1 and 450 itself, is \((1+1)(2+1)(2+1) = 2*3*3 = 18\) factors.
Answer: C
22 Oct 2023, 11:51
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I've revised the question and solution, incorporating additional details for improved clarity. I trust this makes it more comprehensible.
