What portion of the set of unique factors of the product of 24 and 385

Question

What portion of the set of unique factors of the product of 24 and 385 are prime factors?
 
(A)  \frac{5}{7}

(B)  \frac{7}{32}

(C)  \frac{5}{64}

(D)  \frac{6}{64}

(E)  \frac{7}{64}

abhishekdadarwal2009 · Accepted Answer

IMO : C

What portion of the set of unique factors of the product of 24 and 385 are prime factors?

factors of 24=1,2,3,4,6,8,12,24 = 8 factors
factors of 385=1, 5, 7, 11, 35, 55, 77, 385 =8 factors

factors of 24*385 = 8*8=64,

prime factors among all factors

2,3,5,7,11 =5

so solution = prime factors/Total factors

=5/64

firas92 · Answer

24*385 = 2*2*2*3*5*7*11 = 2^3*3^1*5^1*7^1*11^1

So number of factors of the product  24*385 = (3+1)*(1+1)*(1+1)*(1+1)*(1+1) = 64

Out of the 64 factors, 2,3,5,7,11 are the prime factors

So, the portion of the set of unique factors of the product of 24 and 385 are prime factors =  \frac{5}{64}

Kinshook · Answer

What portion of the set of unique factors of the product of 24 and 385 are prime factors?

(A) \frac{5}{7}

(B) \frac{7}{32}

(C) \frac{5}{64}

(D) \frac{6}{64}

(E) \frac{7}{64}

24 = 2^3*3 
385 = 5*7*11
Product of 24 and 385 = 24*385= 2^3*3*5*7*11

# of factors of 24*385 = 4*2*2*2 = 64
# of prime factors = 5 {2,3,5,7 & 11}

Portion of prime factors to total unique factors =  \frac{5}{64}

IMO C

Prasannathawait · Answer

answer has to be C.

total number of factors of 385 and 24 is 64. (by breaking down into primefactors and calculating factors)

Prime numbers in the multiplication are 2 3 5 7 11

So the answer is 5/64

Anki2609 · Answer

Factors for 24 = 1,2,3,4,6,8,12,24
Factors for 385 = 1,5,7,11,35,55,77,385

Total number of prime factors in (24*385)=5 (2,3,5,7,11)
Looking at the options it can only be  option C  =  \frac{5}{64}

JonShukhrat · Answer

What portion of the set of unique factors of the product of  24  and  385  are prime factors?

In oder to reach the correct answer choice faster, we don't actually need to multiply those numbers. We will merely prime factorize them and find the overall number of factors:

24=2^3*3

385=5*7*11

The product of all prime factors will give us the product of the numbers themselves:

24*385=2^3*3*5*7*11

Next, we need to add  1  to the exponents of all prime factors and then multiply the results:

(3+1)*(1+1)*(1+1)*(1+1)*(1+1)=64

So  24*385  has overall  64  distinct factors, including  5  prime factors  2, 3, 5, 7,  and  11 . Thus the portion of  64  unique factors (ratio of prime to all) that are prime is  5/64 .

Hence  C

exc4libur · Answer

Rule: num.factor(x) is equal to the product of the "power+1" of its prime factors

prime factors of (24)=(2ˆ3•3)
prime factors of (385)=(7•11•5)
num.factors(24•385)=(2ˆ3•3•7•11•5)=(3+1)(1+1)(1+1)(1+1)(1+1)=64
no. of primes / total num factors = 5 {2,3,7,11,5} /64

Answer (C).

nick1816 · Answer

What portion of the set of unique factors of the product of 24 and 385 are prime factors?

24*385= 2^3*3*5*7*11
Total number of unique factors= 4*2*2*2*2=64
Prime factors= 5

Probability= 5/64

IMO C

manass · Answer

https://gmatclub.com/forum/download/fil ... 0a1b19362c

Rajeet123 · Answer

Might sound silly, but could you please explain how you got the number of factors as 64? Is that some formula?

firas92 · Answer

Hi  Rajeet123

Sure, remembering this as a formula will definitely be useful. If a number  N  can be expressed as  p1^a*p2^b*p3^c  (prime factorization), where  p1, p2  and  p3  are distinct prime numbers, then the number of factors of  N  is  (a+1)*(b+1)*(c+1)

Consider the example of  72 .

72=2^3*3^2

By our formula, the number of factors of  72  should be  (3+1)*(2+1) = 4*3 = 12

If we need to analyze this by counting methods, in how many ways can we divide  2^3*3^2  such that there is no remainder?

In the denominator of the fraction  \frac{2^3*3^2}{Denominator} , we obviously cannot have any term other than 2 or 3. How many powers can 2 and 3 take?

2  can take powers  0, 1, 2  or  3  -  4  options
 3  can take powers  0, 1  or  2  -  3  options

By fundamental principles of counting, we can just multiply these options to arrive at the number of factors. That is again,  4*3=12

So the factors are 
 2^0*3^0 , 
 2^0*3^1 , 
 2^0*3^2 , 
 2^1*3^0 , 
 2^1*3^1 , 
 2^1*3^2 , 
 2^2*3^0 , 
 2^2*3^1 , 
 2^2*3^2 , 
 2^3*3^0 , 
 2^3*3^1 , 
 2^3*3^2

Hope this is clear!

Basshead · Answer

24 = 2^3 * 3
385 = 5 * 7 * 11

total factors = (2+1)(1+1)(1+1)(1+1)(1+1) = 64

Prime factors = 2, 3, 5, 7, 11

5/64. Answer is C.

Adarsh_24 · Answer

Tricky question. How many are prime factors among unique factors. So even the prime factor count is unique.
I got it wrong as 7/64.

What portion of the set of unique factors of the product of 24 and 385

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What portion of the set of unique factors of the product of 24 and 385

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