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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  [#permalink]

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New post Updated on: 11 Jul 2019, 09:09
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I did have small issues with this, I just got to this part in my study material.

But my process:
Prime Factorization of the two

24 = 2^3, 3^1, so two unique prime factors 2 & 3, total factors = (3+1)(1+1) = 4*2 = 8

so unique prime factors to total factors is 2/8

385 = 5^1, 7^1,11^1, so two unique prime factors 5,7,11 total factors = (1+1)(1+1)(1+1) = 8

so unique prime factors to total factors is 3/8

So 2/8 * 3/8 = 6/64 ,

but I also think it is C, because there is 64 total factors, but if you multiply 2 * 3 you get 6, but the unique prime factors is 2,3,5,7, 11. Reason I selected C.

2 + 3 = 5
8 * 8 = 64
5/64

Originally posted by kapstone1996 on 11 Jul 2019, 08:37.
Last edited by kapstone1996 on 11 Jul 2019, 09:09, edited 1 time in total.
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:37
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IMO C.

24= (2^3) * (3).
385 = 5*11*7.
The product 24*538 can be represented as 2^3 * 3 * 5 * 7 * 11.
The total number of factors would be 4*2*2*2*2 = 64.
Prime factors = 2,3,5,7,11.
Therefore, the portion of the set of unique factors of the product of 24 and 385 which are prime factors are 5/64
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:38
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IMO correct ans is C - Explanation is provided as below
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:44
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Prime factorization of 24*385=2^3*3^1*5^1*7^1*11^1

sample space=no of factors of 2^3*3^1*5^1*7^1*11^1=(3+1)(1+1)(1+1)(1+1)(1+1)=4*2*2*2*2=64

No of favorable events=No of prime factors=5 (2,3,5,7,11)

Hence required probability=5/64

Ans. (C)
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:44
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24X385 = 2^3.3.5.7.11
Total factors = 4X2X2X2X2=64

no of prime factors =5

Answer = 5/64
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What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post Updated on: 12 Jul 2019, 02:12
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

Prime factors of 24 - 2,2,2,3
Factors of 24 - 2, 3, 4, 6, 8, 12, 24

Prime factors of 385 - 5,7,11
Factors of 385 - 5, 7, 35, 55, 77, 385

There are 5 prime factors in all.

Looking at options there are only two with 5 as numerator.
5/7 and 5/64

It cannot be 5/7 since there are clearly more than 7 factors in the product.

Hence it has to be 5/64

Option C

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Originally posted by prashanths on 11 Jul 2019, 08:46.
Last edited by prashanths on 12 Jul 2019, 02:12, edited 1 time in total.
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:49
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IMO-C

24 X 385 = 2^3 * 3 * 5 * 7 * 11
No of factors= 4*2*2*2*2= 64
Prime Factors= 5

Req Fraction= 5/64
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:51
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

(A) 5/7

(B) 7/32

(C) 5/64

(D) 6/64

(E) 7/64

\(24 * 385 = 2^3 * 3^1 * 7 * 11 *5\)

No. of distinct prime factor = 5

Total number of factors = 64

Ans - C
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:51
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

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

So the total amount of factors: (3+1)*(1+1)*(1+1)*(1+1)*(1+1)=4*2^4=64
Unique prime factors are: 2; 3; 5; 7; 11 -> total amount = 5

The portion: 5/64
Answer: C
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:52
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Quote:
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 * 385 = \(2^3 * 3 * 5 *7*11\)
Total factors = 4*2*2*2*2= 64.
There are 5 prime factors.
Option C
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What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post Updated on: 12 Jul 2019, 01:10
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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
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Originally posted by JonShukhrat on 11 Jul 2019, 08:54.
Last edited by JonShukhrat on 12 Jul 2019, 01:10, edited 1 time in total.
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What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post Updated on: 12 Jul 2019, 01:06
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Please refer to the attached below
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Originally posted by mira93 on 11 Jul 2019, 08:54.
Last edited by mira93 on 12 Jul 2019, 01:06, edited 1 time in total.
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:59
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

Solution:
24= 2^3*3 & 385 = 5*7*11
so 24*385 = 2^3*3^1*5^1*7^1*11^1 ==> number of factors in 24*385 = (3+1)*(1+1)*(1+1)*(1+1)*(1+1) = 2^6
And number of prime factors in 24*385 = 5 (prime factors in 24*385 are 2, 3, 5, 7 & 11)
prime_factors/total_factors = 5/64


(A) \(\frac{5}{7}\)

(B) \(\frac{7}{32}\)

(C) \(\frac{5}{64}\) --> correct

(D) \(\frac{6}{64}\)

(E) \(\frac{7}{64}\)
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 08:59
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IMO C

What portion of the set of unique factors of the product of 24 and 385 are prime factors?
We get, 24*385 = 2^3*3*5*7*11
Total no. of prime factors = 5 (2,3,5,7,11)
Total unique factors = multiply (power+1) of each prime factor = (3+1)*2*2*2*2 = 64

Fraction = 5/64
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 09:02
set of unique factors of the product of 24 and 385 are prime factors

24 x 385 = (2X2X2X3)X(5X7X11)

Total Number of factors = 7
Unique number of factors = 5

5/7 Answer A
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 09:03
So fraction of the set of unique factors of the product of 24 and 385 which are prime .

24•385 —> 2^3•3•5•7•11
Total number of unique factors=(3+1)•(1+1)•(1+1)•(1+1)•(1+1) =2^6

Total number of prime factors = 3+1+1+1+1 =7

Hence, fraction is 7/64
Answer E

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What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post Updated on: 12 Jul 2019, 06:37
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

To find the prime factors of the set, we need to break down 24 and 385:
\(24 = 2^3 * 3^1\), and
\(385 = 5^1 * 7^1 * 11^1\)

Therefore, the prime factors are
\(2,3,5,7,11.\)

To find the total number of factors, we calculate the product of the powers of the prime factors,
after adding 1 to each power:

\(2^(^3^) * 3^(^1^) * 5^(^1^) * 7^(^1^) * 11^(^1^)\)
\(2^(^3^+^1^) * 3^(^1^+^1^) * 5^(^1^+^1^) * 7^(^1^+^1^) * 11^(^1^+^1^)\), so

The total number of factors is \(4 * 2 * 2 * 2 * 2 = 64\),

and the proportion of prime factors to total factors is, therefore \(5/64\).

If my assumptions are correct, the answer is C.

Originally posted by Jbrandl1 on 11 Jul 2019, 09:05.
Last edited by Jbrandl1 on 12 Jul 2019, 06:37, edited 3 times in total.
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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 09:09
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We have:
24=3x2^3
385=5x7x11

So the product of:
24x385=(2^3)x3x5x7x11

This product will have (3+1)(1+1)(1+1)(1+1)(1+1)=64 factors

(According to the rule that (p^a)x(q^b) will have (a+1)(b+1) factors)

There are 5 unique prime factors in the product of 24 and 385 (which are 2,3,5,7,11)

So the portion we are looking for is 5/64

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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 09:14
Prime factors: 2-3-5-7-11
All factors: 1-2-3-5-7-11- 24*385

The answer is 5/7 - A

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Re: What portion of the set of unique factors of the product of 24 and 385  [#permalink]

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New post 11 Jul 2019, 09:15
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The product of 24 and 385: (24*385) = (2^3 *3)*(5*7*11)

Total number of factors of (24*385) is (3+1) * (1+1) * (1+1) * (1+1) * (1+1) = 4*2*2*2*2 = 64 nos.
Prime factors of (24*385) = 2,3,5,7,11 (5 nos)
Portion of the set of unique factors of the product of 24 and 385 are prime factors = 5/64

Answer is (C): 5/64
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Re: What portion of the set of unique factors of the product of 24 and 385   [#permalink] 11 Jul 2019, 09:15

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