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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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11 Jul 2019, 08:00
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35% (medium)

Question Stats:

75% (01:37) correct 25% (02:04) wrong based on 287 sessions

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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) $$\frac{7}{64}$$

 This question was provided by Veritas Prep for the Game of Timers Competition

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

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Updated on: 21 Jul 2019, 11:35
2
$$24*385 = 2*2*2*3*5*7*11 = 2^3*3^1*5^1*7^1*11^1$$

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 = $$\frac{5}{64}$$

Originally posted by firas92 on 11 Jul 2019, 08:09.
Last edited by firas92 on 21 Jul 2019, 11:35, edited 2 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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11 Jul 2019, 08:07
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totoal number of factors of: 2^3*3*5*7*11 (note the product can be written as this) = (3+1)*2*2*2*2 = 64
total prime factors = 5
ratio is 5/64
so 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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Updated on: 11 Jul 2019, 10:41
1
product of 24 and 385

24*385 = $$2^3$$*3*5*7*11
total number of factors = (3+1)(1+1)(1+1)(1+1)(1+1) = 4*2*2*2*2 = 64
total number of prime factors = 2,3,5,7,11 = 5

portion that are prime factors = $$\frac{5}{64}$$

Originally posted by shridhar786 on 11 Jul 2019, 08:11.
Last edited by shridhar786 on 11 Jul 2019, 10:41, 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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11 Jul 2019, 08:16
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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
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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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11 Jul 2019, 08:16
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24= 2^3 * 3
385= 5*7*11
Product= 2^3 * 3 * 5 * 7 * 11

Total factors of their products-> (3+1) * (1+1) * (1+1)* (1+1)* (1+1) = 4*16= 64
Primes= 2, 3, 5, 7 and 11.

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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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11 Jul 2019, 08:17
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24= 2^3 * 3. Total possible factors = (3+1)*(1+1) = 8
385=5*7*11. Total possible factors =2*2*2=8

Total possible factors of 24 and 385 = 8*8=64

Unique prime factors= 2,3,5,7,11 = 5

Hence 5/64. Option 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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11 Jul 2019, 08:22
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in the product of 24 and 385 prime factorisation gives

$$2^3$$ * 3 * 5 * 11*7

so total factors are 4 * 2*2*2*2 which gives 64 unique factors

out of which only 5 are prime factors 2,3,5,7 and 11

so the ans is 5/64 ie 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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11 Jul 2019, 08:23
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C

Factorize the product of 24 * 385 we get 2^3*3*5*7*11

So total prime factors are 5, and total no of factors are (3+1)*(1+1)^4 = 64

Total portion of prime factors : 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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11 Jul 2019, 08:24
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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

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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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11 Jul 2019, 08:24
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Prime factorization of 24*385=2^3*3*7*11*5
Total factors = 4*2*2*2*2 = 64 (using the method for factors - multiple of each prime's power +1)
Total Primes = 2+3+11+7+5=5

So portion of unique factors - 5/64

IMO 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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11 Jul 2019, 08:25
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

total factors of 24*385 ;2^3*3*5*7*11 ; 4*2*2*2*2 ; 64
total prime factors; 2,3,5,7,11 ; 5
5/64
IMO 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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11 Jul 2019, 08:26
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

This question is very easy. To solve it all you need to know is that the total number of factors of a product is the product of the individual factors.
Total number of factors of the product = 24=> 2^3*3^1 => 4*2 => 8 . & for 385 => 11^1*5^1*7^1 => 2*2*2 = 8 so total is 8*8 = 64

Now for prime numbers = 2, 3,7,5,11 so total is 5

(A) 5/7

(B) 7/32

(C) 5/64

(D) 6/64

(E) 7/64

Hence 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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11 Jul 2019, 08:27
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Answer is c, 64 total factors and 5 is prime factors are 5.

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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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11 Jul 2019, 08:28
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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
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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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11 Jul 2019, 08:29
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24 * 385 = 2^3 * 3^1 * 5^1 * 7^1 * 11^1
Total factors = 64
Prime factors = 5
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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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11 Jul 2019, 08:29
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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 = $$\frac{5}{64}$$
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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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11 Jul 2019, 08:30
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Product 0f 24 * 385, breaking in prime factors:
=> $$2^3$$ * 3 * 5 * 7 *11
Total number of factors => 4 * 2 * 2 * 2 * 2 => 64 (as $$a^2$$ will have (2+1) factors)
Total number of unique prime factors = 2,3,5,7,11 => 5

So portion => 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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11 Jul 2019, 08:32
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

Total factors: 2^3*3*5*7*11
(3+1)(1+1)(1+1)(1+1)(1+1)=64

5 unique prime factors - 2,3,5,7,11

Therefore the 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]

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11 Jul 2019, 08:33
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Perform prime factorisation to get the individual factors:
24 - 2*2*2*3
385 - 5*7*11
Now,
Total number of factors - 64
Total number of unique prime factors -5
hence, 5/64 D
Re: What portion of the set of unique factors of the product of 24 and 385   [#permalink] 11 Jul 2019, 08:33

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