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If n has 15 positive divisors, inclusive of 1 and n, then wh

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If n has 15 positive divisors, inclusive of 1 and n, then wh  [#permalink]

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New post Updated on: 16 Jun 2013, 00:35
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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

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Originally posted by fozzzy on 16 Jun 2013, 00:32.
Last edited by Bunuel on 16 Jun 2013, 00:35, edited 1 time in total.
Edited the question and the tags.
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh  [#permalink]

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New post 16 Jun 2013, 00:58
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9
fozzzy wrote:
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


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.
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh  [#permalink]

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New post 16 Jun 2013, 01:01
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1
fozzzy wrote:
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


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.
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh  [#permalink]

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New post 16 Jun 2013, 01:03
I didn't notice the case of 16 before great explanation. How would you rate this problem on close to 650 or above that?
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh  [#permalink]

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New post 16 Jun 2013, 01:04
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh  [#permalink]

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New post 18 Aug 2017, 05:54
I worked on the problem using the concept of finding number of positive factors of a given number.
If a given number N can be written in its prime factorized form as
N = \(a^{p}\)*\(b^{q}\)*\(c^{r}\).....,
then number of positive factors of a given number N = (p+1)(q+1)(r+1).....
so if number of factors of N = 15, then 15 can be written as
Case 1: (14+1). Thus here p = 14, i.e. N = \(3^{14}\). 3N = 3*\(3^{14}\) = \(3^{15}\) and will have 15+1 = 16 factors
Case 2: 3*5. Thus here p = 2 and q =4. N = \(a^{2}\)*\(b^{4}\). Thus 3N = \(3^{1}\)*\(a^{2}\)*\(b^{4}\), having number of factors as (1+1)(2+1)(4+1) = 30
or N = \(3^{2}\)*\(b^{4}\), thus 3N = \(3^{3}\)*\(b^{4}\), having number of factors as (3+1)(4+1) = 20
or N = \(a^{2}\)*\(3^{4}\), thus 3N = \(a^{2}\)*\(3^{5}\), having number of factors as (2+1)(5+1) = 18

Only 20 and 30 are satisfying thus answer is Option B
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh &nbs [#permalink] 18 Aug 2017, 05:54
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