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# Find the smallest number that has exactly 12 factors.

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Retired Moderator
Status: The best is yet to come.....
Joined: 10 Mar 2013
Posts: 502
Find the smallest number that has exactly 12 factors.  [#permalink]

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20 Oct 2017, 08:48
1
3
00:00

Difficulty:

35% (medium)

Question Stats:

57% (00:35) correct 43% (01:12) wrong based on 110 sessions

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Find the smallest number that has exactly 12 factors.

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

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Hasan Mahmud

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Joined: 21 Aug 2013
Posts: 1412
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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1
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

Math Expert
Joined: 02 Aug 2009
Posts: 7108
Re: Find the smallest number that has exactly 12 factors.  [#permalink]

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20 Oct 2017, 22:05
1
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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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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

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17 Nov 2018, 02:14
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Re: Find the smallest number that has exactly 12 factors. &nbs [#permalink] 17 Nov 2018, 02:14
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