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B
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Difficulty:
75%
(hard)
Question Stats:
58% (02:16) correct
42%
(02:04)
wrong
based on 533
sessions
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If n has 15 positive divisors, inclusive of 1 and n, then which of the following could be the number of divisors of 3n?
I. 20
II. 30
III. 40
A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III only
I. 20
II. 30
III. 40
A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III only
16 Jun 2013, 00:58
Kudos
Bookmarks
fozzzy
n has 15 positive divisors --> \(n=p^{14}\) (the # of factors (14+1)=15) or \(n=p^2q^4\) (the # of factors (2+1)(4+1)=15).
If neither p nor q is 3, then:
\(3n=3p^{14}\) will have (1+1)(14+1)=30 factors.
\(3n=3p^2q^4\) will have (1+1)(2+1)(4+1)=30 factors.
If p=3, then:
\(3n=3p^{14}=3^{15}\) will have (15+1)=16 factors.
\(3n=3p^2q^4=3^3p^4\) will have (3+1)(4+1)=20 factors.
If q=3, then:
\(3n=3p^2q^4=p^23^5\) will have (2+1)(5+1)=18 factors.
Answer: B.
General Discussion
16 Jun 2013, 01:01
Kudos
Bookmarks
fozzzy
Finding the Number of Factors of an Integer
First make 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 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: \(450=2^1*3^2*5^2\)
Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.
For more check here: math-number-theory-88376.html
Hope it helps.
