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

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Director
Joined: 19 Oct 2018
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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, 18:11
1
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
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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, 18:37
1
The number of unique factors for 24*85 are

24*85=2^3*3*5*7*11
So (3+1)*(1+1)*(1+1)*(1+1)*(1+1)
=64

And out of then prime are 2,3,5,7,11

Therefore 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, 19:06
1
What portion of the set of unique factors of the product of 24 and 385 are prime factors?

If we decompose 24*385 into prime factors we get: 2^3 * 3 * 5 *7 *11.

so we have 5 prime factors and # of factors is (3+1)*(2)*(2)*(2)*(2)= 2^6 = 64

so the result is 5/64, (C) is our answer.
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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, 19:17
1
$$24*385 = 2^33^15^17^111^1$$,
so number of unique factors $$= 4*2*2*2*2 = 64$$
and the prime factors involved are $$2,3,5,7,11$$ ,i.e $$5$$ primes

so the portion = $$\frac{5}{64}$$
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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, 20:12
1
factors of the product

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 10 | 11 | 12 | 14 | 15 | 20 | 21 | 22 | 24 | 28 | 30 | 33 | 35 | 40 | 42 | 44 | 55 | 56 | 60 | 66 | 70 | 77 | 84 | 88 | 105 | 110 | 120 | 132 | 140 | 154 | 165 | 168 | 210 | 220 | 231 | 264 | 280 | 308 | 330 | 385 | 420 | 440 | 462 | 616 | 660 | 770 | 840 | 924 | 1155 | 1320 | 1540 | 1848 | 2310 | 3080 | 4620 | 9240

5 distinct prime numbers

so 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, 20:37
1
24=$$2^3$$*3. Overall factors are 4*2=8
385=5*7*11. Overall factors are 2*2*2=8
So product of 24*385 has 64 factors (8*8)
Unique factors are 2,3,5,7,11 (5 in total) - 5/64 (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, 21:09
1
24 and 385
for the product is 24 and 385 = 8*3* 5*7*11 =$$2^3∗3∗5∗7∗11$$
hence total number of prime factors =$$(3+1)∗2∗2∗2∗2$$= 64
but total number of prime factors will be 2,3,5,7,11 = 5
hence fraction =$$\frac{5}{64}$$
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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, 21:34
1
Number of factors of product of 24 and 385(2^3*3^1*7*11*5) = 8*8=64
Prime factors = 2,3,5,7,11
Hence 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, 21:43
1
Product of 24 and 385 = 9240
$$9420 = 2^3 * 3 * 5 * 7 * 11$$

Number of Prime factors = 5
Number of unique Positive Factors = (3 + 1) * (1 + 1) * (1 + 1) * (1 + 1) * (1 + 1) = 64

Therefore, the portion we are looking for is $$\frac{5}{64}$$

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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, 21:47
1
What portion of the set of unique factors of the product of 24 and 385 are prime factors?

here, 24= 2^3 * 3
385= 5*7*11
there are 5 prime factor
total factor = (3+1)*(1+1)*(1+1)*(1+1)*(1+1) =64

So, the correct answer choice is (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, 22:01
1
Question is asking for the fraction of the unique set of factors of 24 * 385. That is, # of unique factors divided by total # of factors.

First, find the prime factorization of the product of 24 and 385.

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

Total # of factors of any number with prime factorization, x^a*y^b*z^c = (a+1)(b+1)(c+1)

From above Total # of factors of the product 24 * 385 = 4*2*2*2*2 = 64

There are 5 unique factors of this product. They are 2, 3, 5, 7, 11.

Hence the portion of the set of unique factors of 24 * 385 = 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, 22:28
1
rephrase the question
x portion of Y factors of 24*285 are prime factors..
x = no of prime factors in the product / total unique factors in 24*385

24 = 2^3 * 3^1 = 8 total factors - unique prime factors are 2, 3
385 = 5^1 * 7^1 * 11^1 = 8 total factors - unique prime factors 5, 3, 7

combined there are 5 unique prime factors
total factors - 8 * 8
portion 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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11 Jul 2019, 22:57
1
product of 24 and 385 = 24x385 --> (2^3x3) x (5x7x11) --> total unique factors are = (3+1)(1+1)(1+1)(1+1)(1+1) = 64, prime factors = 2,3,5,7, and 11, so the answer is 5/64 (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, 23:30
1
Attachments

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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, 23:47
1
24 = 2^3 * 3

385 = 5 * 7 * 11

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

Total no. of factors of 24 * 385 = (3+1) * (1+1) * (1+1) * (1+1) * (1+1) = 64

Prime factors of the product are 2, 3, 5, 7, and 11 ie, no of prime factors = 05

Hence, proportion needed = 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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Updated on: 12 Jul 2019, 08:25
What portion of the set of unique factors of the product of 24 and 385 are prime factors?

Product can be re-written as = 24*385
= 6*4 *77*5
= 2*3*2*2*11*7*5
=$$2^3 *3^1*5^1* 7^1* 11^1$$
So total number of unique factor = (3+1)*(1+1)*(1+1)*(1+1)*(1+1)
= 4*2*2*2*=4*16 = 64

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

So 5 of 64 are prime factors = 5/64 = Option C

Originally posted by baru on 11 Jul 2019, 23:56.
Last edited by baru on 12 Jul 2019, 08:25, 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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Updated on: 12 Jul 2019, 10:01
Soln:

prime factors of 24 :2,2,2,3
Prime factors of 385: 5,7,11

Product of 24 and 385 will have total prime factors:(3+1) (1+1) (1+1)(1+1)(1+1)
TOTAL COUNT 64
unique prime factors : 2,3,5,7,11 ( Total count 5)

So portion of the set of unique factors of the product of 24 and 385 are prime factors 5/64

CORRECT ANS C

Originally posted by komals06 on 12 Jul 2019, 00:08.
Last edited by komals06 on 12 Jul 2019, 10:01, 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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12 Jul 2019, 01:14
1
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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12 Jul 2019, 01:22
1
What portion of the set of unique factors of the product of 24 and 385 are prime factors?

(A)
5
7

57

(B)
7
32

732

(C)
5
64

564

(D)
6
64

664

(E)
7
64

764

Solutions:
Factors of 24=$$2^3$$ *3
Factors of 385=5 * 7* 11
Unique Factors=2 * 3 *5 * 7 * 11
Number of unique factors =5
Total number of prime factors =(3+1)(1+1)((1+1)(1+1)(1+1)=4*2*2*2*2=64
The portion of set of unique factors of the product of 24 and 385=$$\frac{5}{64}$$
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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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12 Jul 2019, 02:15
1
prime factorization of $$24*385= 2^3*3*5*7*11$$

Total number of factors $$= (3+1)*(1+1)*(1+1)*(1+1)*(1+1)= 64$$
Distinct prime factors $$= (2,3,5,7,11)= 5$$
Required ratio $$=\frac{5}{64} .$$
IMO C
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Re: What portion of the set of unique factors of the product of 24 and 385   [#permalink] 12 Jul 2019, 02:15

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