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Updated on: 26 Oct 2023, 09:53
Originally posted by suntaurian on 28 Feb 2008, 09:23.
Last edited by Bunuel on 26 Oct 2023, 09:53, edited 2 times in total.
Last edited by Bunuel on 26 Oct 2023, 09:53, edited 2 times in total.
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D
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For function \(\#\), \(\#x\) represents the number of distinct positive divisors of the positive integer \(x\). What is the value of \(\#(\#90)\)?
A. 3
B. 4
C. 5
D. 6
E. 7
M01-35
A. 3
B. 4
C. 5
D. 6
E. 7
M01-35
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27 Sep 2014, 06:00
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For function \(\#\), \(\#x\) represents the number of distinct positive divisors of the positive integer \(x\). What is the value of \(\#(\#90)\)?
A. 3
B. 4
C. 5
D. 6
E. 7
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.
Back to the original question:
The question defines \(\#x\) as the number of distinct positive divisors of \(x\). For example, \(\#6=4\), as 6 has 4 distinct positive divisors: 1, 2, 3, and 6.
Question: \(\#(\#90)=\)?
Since \(90 = 2 * 3^2 * 5\), the number of factors of 90 is: \((1+1)(2+1)(1+1) = 12\). So \(\#90 = 12\). Next, we need to find \(\#(\#90) = \#12\). Now, as \(12 = 2^2 * 3\), the number of factors of 12 is: \((2+1)(1+1) = 6\).
Answer: D
A. 3
B. 4
C. 5
D. 6
E. 7
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.
Back to the original question:
The question defines \(\#x\) as the number of distinct positive divisors of \(x\). For example, \(\#6=4\), as 6 has 4 distinct positive divisors: 1, 2, 3, and 6.
Question: \(\#(\#90)=\)?
Since \(90 = 2 * 3^2 * 5\), the number of factors of 90 is: \((1+1)(2+1)(1+1) = 12\). So \(\#90 = 12\). Next, we need to find \(\#(\#90) = \#12\). Now, as \(12 = 2^2 * 3\), the number of factors of 12 is: \((2+1)(1+1) = 6\).
Answer: D
General Discussion
rey1009
Joined: 20 Oct 2023
Last visit: 31 Jul 2024
Posts: 16
Location: India
Concentration: Technology, Entrepreneurship
Schools: Cornell '26 Tepper '26 NUS '26
GPA: 3.74
WE:Engineering (Computer Software)
26 Oct 2023, 11:12
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#x= the number of distinct positive divisors of the positive integer x
#(#90)
positive divisors of 90 = 1, 2,3,5,6,9,10,15,18,30,45,90
#90 =12
#12 = 1, 2, 3, 4, 6, 12 => 6
#(#90)
positive divisors of 90 = 1, 2,3,5,6,9,10,15,18,30,45,90
#90 =12
#12 = 1, 2, 3, 4, 6, 12 => 6
