What is the probability that a randomly selected positive factor of 72

Question

What is the probability that a randomly selected positive factor of 72 is a multiple of 6?

(A)  \frac{1}{4}

(B)  \frac{1}{3}

(C)  \frac{2}{5}

(D)  \frac{1}{2}

(E)  \frac{3}{5}

GMATinsight · Answer

72 = 2^3*3^2

Factors of 72 = (3+1)*(2+1) = 12

72/6 = 12

12 = 2^2*3

Factors of 12 = (2+1)*(1+1) = 6

i.e. Factors of 72 that are divisible by 6 = 6
Factors of 72 = 12

Required probability = 6/12 = 1/2

Answer: Option D

ArunSharma12 · Answer

72 = 2 ^3* 3^2 
total number of factors = 12
number of factors that are multiple of 6 =  6 *(2^2*3)  = 3*2 = 6
Probability =  \frac{1}{2}

Archit3110 · Answer

factors of 72 ; 2^3*3^2
total factors ; 4*3 ; 12
we have is 2*2*2 * 3*3 
so multiples of 6 i.e (3*2) would be 3c1*2c1 ; 6 ways
we get ; 6/12 ; 1/2

IMO D

GMATinsight · Answer

I don't see that combinatorics you have used here is making sense.

Factors will be multiple of 6 when we have minimum 3^1 and 2^1

so while counting factors of 72 we will ignore zero power
For factors calculation of 72 we take total combinations of  {2^0, 2^1, 2^2, 2^3} {3^0, 3^1, 3^2}

But when we have to make them multiple of 6 then we will make combinations of  { 2^0 , 2^1, 2^2, 2^3} { 3^0 , 3^1, 3^2}

now, total combinations = 3*2 = 6

If this is what you meant with 3c1 and 2c1 then you are correct

Archit3110

chetan2u · Answer

Ways to do it..
 72=2^3*3^2

I. 
We can take out 6 out of this and find remaining factors, which will be multiple of 6
Number of factors of 72 72=2^3*3^2 ...(3+1)(2+1)=4*3=12
Number of multiples of 6 as factors of 72 \frac{72}{6}=12=2^2*3^1 , so factors of 12 = (2+1)(1+1)=3*2=6
Probability =  \frac{6}{12}=\frac{1}{2}

OR

II. 
When will a factor of  72=2^3*3^2  not contain 6 in it?:  when the factor does not contain both 2 and 3 together.. 
So Only factors with 3s..... 3^0, 3^1, 3^2 ...so 3 factors
Only factors with 2s.....  2^1, 2^2, 2^3 ....so 3 factors, and we do not count 2^0 as it is same as 3^0 or 1
so 3+3=6 in total.
Total factors of 72 are 12 as seen above..
Probability =  \frac{6}{12}=\frac{1}{2}

D

Archit3110 · Answer

GMATinsight ; yes I meant the same by using 3c1*2c1 ( highlighted ) ... thanks

rsmalan · Answer

Hello sjuniv32

This is a great Problem Solving question testing concepts in Probability and Factorisation.
Let's begin

Question stem: Amongst the factors for the number 72, we are required to find the probability of selecting a factor divisible by 6
Probability(randomly selected positive factor of 72 is a multiple of 6) =  \frac{ factors of 72 that are multiples of 6 }{ total number of factors of 72} 
 72 = {2^3 * 3^2}

So, total number of factors of 72 = (3+1).(2+1) = 4 x 3 = 12

Factors of 72 divisible by 6 are: 6, 12, 18, 24, 36, 72 i.e. total of 6 items

Thus, Probability(randomly selected positive factor of 72 is a multiple of 6) =  \frac{factors of 72 that are multiples of 6 }{ total number of factors of 72} 
So, we have  \frac{ 6 }{ 12 }= \frac{1 }{ 2 }

Answer (D)

Pls share your thoughts. Is their anything you would like to be better explained in the above solution?
Best regards

chetan2u · Answer

Hi we require a 6, so we take out a 6 from 72, and then find factors of it. 
Here we had 1,2,3,4,6,12
So factors that are multiple of 6 are 1*6, 2*6,3*6,4*6,6*6,12*6

Turjoy98 · Answer

generis  SIR, I'm not sure of the problem above. Is it a probability PS or Number property?

GMATinsight · Answer

Hi Turjoy98 This problem is already tagged as both Probability problem as well as Number properties problem in which we learn to find the specific types of factors of any Number.

However, Solving this problem in topic probability before one has done the topic Number property will NOT be advisable.

xiaxyl · Answer

Can this problem be explained in a simpler manner?Unable to understand it

Bunuel · Answer

What is the probability that a randomly selected positive factor of 72 is a multiple of 6? (A) \frac{1}{4} (B) \frac{1}{3} (C) \frac{2}{5} (D) \frac{1}{2} (E) \frac{3}{5} Calculate the total number of positive factors of 72: 72 = 2^3 * 3^2 Thus, the total number of positive factors of 72 is (3 + 1)(2 + 1) = 12 (check here: https://gmatclub.com/forum/math-number- ... 88376.html). 72 is a relatively small number, so it's easier to simply list all its factors that are multiples of 6: (2 * 3) = 6, (2 * 3) * 2 = 12, (2 * 3) * 3 = 18, (2 * 3) * 2^2 = 24, (2 * 3) * 2 * 3 = 36, and 72 itself. So, a total of six multiples of 6. Thus, the probability is 6/12 = 1/2. Answer: D.

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What is the probability that a randomly selected positive factor of 72

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What is the probability that a randomly selected positive factor of 72

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