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

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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, 09:16
2
Quote:
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

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

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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, 09:26
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Prime factorisation of $$24*385 = 2^3 * 3 *5*7*11$$

Total number of factors = $$4*2*2*2*2 = 64$$

Number of prime factors = 5

So fraction 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, 09:32
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To solve for the unique prime factors of the product of 24 and 385. The factors are as follows:

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

multiplying the 24 and 385 will generate the following factors; 1,2,3,4,5,6,7,8,9,10,11, and so on.

Thus the unique prime factors of the product of 24 and 385 are 2,3,5,7, and 11.

Only answer choice with 5 as numerator is "A" and "C".

Since the unique factors of the product is more than 7

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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, 10:12
1
The answer is 5/64 here is why

24 = 2^3 * 3. (2 unique primes)

385 = 5 * 7 * 11 (3 unique primes)

Total number of factors: 4 * 2 * 2 * 2 * 2 ( take exponents of prime factorization add one to each them multiply to get total number of factors). This yields 8 *8 = 64

You get 5/64

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Originally posted by Peter2719 on 11 Jul 2019, 09:51.
Last edited by Peter2719 on 11 Jul 2019, 10:12, 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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11 Jul 2019, 09:53
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?
breakdown the product into prime factors; 2^3 * 3 * 5 * 7 * 11; (2,3,5,7,11 -> 5 unique factors)
in total there are (3+1)(2)(2)(2)(2)= 64 unique factors ;

ans: 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, 09:55
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Working on the stem:
$$24*385=2^3*3*5*7*11$$

We need to find: number of prime factors/total number of positive factors

number of prime factors: 5
Total number of factors: 4*2*2*2*2=64

IMO
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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11 Jul 2019, 09:55
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

We need to find:

Numerator = UNIQUE PRIME FACTORS of (24x385)

Denominator = UNIQUE FACTORS of (24x385)

Start by finding prime factors of 24 and 385

24 = 2x2x2x3

385 = 5x7x11

So Prime factors of (24x385) = (2^3)*3*5*7*11

Unique prime foctors = 2,3,5,7,11..... NUMERATOR = 5

To get unique factors of a number, we need to add 1 to the power of each prime factors and multiply the resulting numbers. 2 has power 3 and others have power 1:

so, number of unique factors = (3+1)*2*2*2*2 = 64 = DENOMINATOR

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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, 10:00
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Prime factorisation of 24 is 2^3 X3, so distinct factors are 8
Prime factorisation of 385 is 5x7x11, so distinct factors are 8

Number of unique factors of product of 24 and 385 are 8x8 = 64.
Out of them prime factors are 2,3,5,7,11. Total number of prime factors is 5

5 out of 64 factors are prime factors.

C is correct.
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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, 10:22
1
12*385 can be written as

2^3*3*11*7*5

No.of factors are 4*2*2*2*2=64

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

So answer is 5/64 that 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, 10:26
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

24 can be written as 2X2X2X3 = 2 unique factor
385 can be written as 11X7X5 = 3 unique factor

24X385 = 5 unique prime factor and 64 total factor

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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, 10:44
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IMO C
solving we get 2^3 *3*5*7*11, so total of 5 prime factors, with no of factors 3+1 *2*2*2*2= 64, 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, 11:15
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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) $$\frac{7}{64}$$

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

So the number of unique factors= (3+1) * (1+1) *(1+1) * (1+1)= 64

Number of prime factors= 5 ------> portion of the set of unique factors of the product of 24 and 385 are prime factors = $$5/64$$

IMO. the correct answer 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, 11:39
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24*385=$$2^{3}$$*3*5*7*11

Total Number of factors of 24*385=4*2*2*2*2=64
Number of unique prime factors=5 {2,3,5,7,11}

Required ratio=$$\frac{5}{64}$$
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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11 Jul 2019, 12:19
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total factors of 24 and 385 = 64
Different prime factors of 24 and 385 is 2,3,5,7,11
portion = 5/64
Hence ans 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, 12:29
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

Our answer would be : $$\frac{unique factors 24*385}{total factors of 24*385}$$

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

So, unique factors are : 2, 3, 5, 7, 11 => Total number = 5
Total number of factors [2*4] * [2*2*2] = 64 (Power of each prime factor+1 & multiplication)

So, our answer should be (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, 13:19
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What portion of the set of unique factors of the product of 24 and 385 are prime factors?

total primes numbers are 5 and the total set of numbers would be
24 = 2^3*3
factors of 24= (3+1)*(1+1)= 8

385 = 5*11*7
factors of 385 = (1+1)*(1+1)*(1+1)= 8

Product of 24 * 385
so factors will be 8*8 = 64

prime numbers / 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, 13:31
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Prime factorization of 24*385 is (2^3)*3*5*7*11
But the number of unique factors of a number which has a prime factorization of (a^x)*(b^y) where a and b are prime numbers and x and y are integers equals (x+1)*(y+1).
Hence the unique factors of 24*385=(4)(2)(2)(2)(2)=64.
There are 5 prime factors,
Hence portion of unique factors which are prime =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, 13:38
Total factors of:
24- 2*2*2*3
385- 5*7*11

Therefore:fav outcome/ Total outcome
==>5/7
IMO-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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11 Jul 2019, 13:43
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only 2,3, 5, 7 & 11 are prime factors of 9240 out of 64. (C) is the 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, 16:52
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24=2^3 *3
385=5*7*11

Total no of factors in it's products =(3+1)(1+1)(1+1)(1+1)(1+1)= 64
Total prime factors=5

Ratio =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, 16:52

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