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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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15 Jun 2013, 23:32
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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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 15 Jun 2013, 23:35, edited 1 time in total.
Edited the question and the tags.
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Joined: 02 Sep 2009
Posts: 43831
Re: If n has 15 positive divisors, inclusive of 1 and n, then wh [#permalink]

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15 Jun 2013, 23:58
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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.

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

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16 Jun 2013, 00:01
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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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Posts: 854
Re: If n has 15 positive divisors, inclusive of 1 and n, then wh [#permalink]

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16 Jun 2013, 00:03
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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Math Expert
Joined: 02 Sep 2009
Posts: 43831
Re: If n has 15 positive divisors, inclusive of 1 and n, then wh [#permalink]

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16 Jun 2013, 00:04
fozzzy wrote:
I didn't notice the case of 16 before great explanation. How would you rate this problem on close to 650 or above that?

Well, ~650 I guess.
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Re: If n has 15 positive divisors, inclusive of 1 and n, then wh [#permalink]

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18 Aug 2017, 04:54
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   [#permalink] 18 Aug 2017, 04:54
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