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# Find the smallest number that has exactly 12 factors.

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Retired Moderator
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Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 09:48
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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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Joined: 22 Aug 2013
Posts: 893
Location: India
Re: Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 11:00
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.

Lets start with option E (smallest number): 60.
60 = 2^2 * 3 * 5. No of factors = 3*2*2 = 12.
So this has 12 factors, no need to check further.

Hence answer is E.
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Find the smallest number that has exactly 12 factors. [#permalink]

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20 Oct 2017, 12:10
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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

Because the prompt asks for the smallest number with 12 factors, start with 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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Formerly genxer123

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

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20 Oct 2017, 23:05
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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 $$N = 2^2*3^1*5^1=4*3*5=60$$
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Re: Find the smallest number that has exactly 12 factors.   [#permalink] 20 Oct 2017, 23:05
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# Find the smallest number that has exactly 12 factors.

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