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# Find the smallest number that has exactly 12 factors.

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CR Forum Moderator
Status: The best is yet to come.....
Joined: 10 Mar 2013
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Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 08:48
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Find the smallest number that has exactly 12 factors.

A. 2048
B. 486
C. 96
D. 72
E. 60
[Reveal] Spoiler: OA

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Hasan Mahmud

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Location: India
Re: Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 10:00
Best way in this is to just start with the smallest number in the option and check whether it has 12 factors or not. This way probably answer will be arrived at fastest.

60 = 2^2 * 3 * 5. No of factors = 3*2*2 = 12.
So this has 12 factors, no need to check further.

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Joined: 22 May 2016
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Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 11:10
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Mahmud6 wrote:
Find the smallest number that has exactly 12 factors.

A. 2048
B. 486
C. 96
D. 72
E. 60

1) list factors of 60 in pairs (answers C, D, and E are doable)

1 x 60
2 x 30
3 x 20
4 x 15
5 x 12
6 x 10

That's 12. Correct

2) Prime factor / exponent method

a) Prime factorize 60:

$$2^23^15^1$$

b) take each prime's exponent and add 1 (e.g., 3 has exponent of 1):

(2 + 1) = 3
(1 + 1) = 2
(1 + 1) = 2

c) Multiply those numbers. Result = total # of factors of 60, including 60 and 1

(3 * 2 * 2) = 12

60 has 12 factors

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Math Expert
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Posts: 5347

Kudos [?]: 6125 [1], given: 121

Re: Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 22:05
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Expert's post
gvij2017 wrote:
why not 72?
72= 2^3*3^2
2 and 3 are prime no. Total no of factors: (1+3)*(1+2)= 4*3=12
D is equally correct choice. Plz correct if I am wrong.

Hi...

we are looking for the smallest and 60<72 so answer is 60..

But the proper method would be, although best would be to test choices starting from lowest

$$12=3*2*2=(2+1)(1+1)(1+1)$$... so factors are to power of 2,1 and 1
Number = $$N = a^2*b^1*c^1$$... take minimum value of a, then b and then c...
so $$N = 2^2*3^1*5^1=4*3*5=60$$
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 6125 [1], given: 121

Re: Find the smallest number that has exactly 12 factors.   [#permalink] 20 Oct 2017, 22:05
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