What is the smallest positive integer that has exactly 12 positive fac

Question

What is the smallest positive integer that has exactly 12 positive factors?

A. 60
B. 120
C. 144
D. 211
E. None of the above

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GMATinsight · Answer

If  N = a^p*b^q*c^r... 
where a, b, c... are distinct primes
Number of factors of  N = (p+1)*(q+1)*(r+1)*...

STRATEGY : 
-For smallest number the Prime number i.e. a, b, c etc must be as small as possible and 
- Exponents of Primes should be broken into parts which makes the number smallest
- Bigger prime number must have smaller power to keep number as small as possible

12 = 6*2 = (5+1)*(1+1)

i.e. Number may be  2^5*3^1 = 96

OR

12 = 4*3 = (3+1)*(2+1)

i.e. Number may be  2^3*3^2 = 72

OR

12 = 2*2*3 = (1+1)*(1+1)*(2+1)

i.e. Number may be  2^2*3*5 = 60

i.e. smallest such number must be 60

Answer: Oprton A

krishnav89 · Answer

Prime factorize all four options while starting itself for 60 = 2^2*3^1*5^1 i got the number of positive factor as 12.
using Number of factors of N=(p+1)∗(q+1)∗(r+1)∗...

(2+1)*(1+1)* (1+1) = 12.

fmcgee1 · Answer

I just looked for all the factor pairs of the numbers, starting with 60.

Factor pairs of 60:
1 * 60
2 * 30
3 * 20
4 * 15
5 * 12
6 * 10

And that's it...answer is A.

Fdambro294 · Answer

Concept: When we are Given the Total Count of Factors of a Number and need to decide the MINIMUM Value that Number can take, the goal should be to try and perform "Maximum Splitting" with the Prime Factors ---- b/c Powers grow Exponentially very Quickly

What I mean is, if the Total Factors of a Number = 12

then understanding the Logic behind the Find the Total Factors Rule, you should look to try and "SPLIT" the Powers of the Prime Bases up as much as you can.

If All the Variables are Prime Numbers, for instance the Number can be:

a^11

or

a^5 * b^1

or

a^3 * b^2 ----- Total Factors = (3 + 1) * (2 + 1) = 4 * 3 = 12 Total Factors

or

a^1 * b^1 * c^2 ---- Total Factors = (1 + 1) * (1 + 1) * (2 + 1) = 12 Total Factors

The Last Case in which the Powers are "SPLIT" between the Prime Bases as much as possible is likely the Minimum Value we can Find that will have 12 Factors.

Assign the Smallest Prime Factor of 2 to Variable c.

2^2 * 3^1 * 5^1 = 4 * 3 * 5 = 60

This is the Minimum Value we can find that has 12 Factors

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What is the smallest positive integer that has exactly 12 positive fac

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What is the smallest positive integer that has exactly 12 positive fac

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