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How many factors of 240 are also multiples of 3 ?

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gmatquant25
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How many factors of 240 are also multiples of 3 ? [#permalink] New post   23 Sep 2014, 10:19
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How many factors of 240 are also multiples of 3 ?

A. 5
B. 8
C. 9
D. 10
E. 20
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D
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Bunuel
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How many factors of 240 are also multiples of 3 ? [#permalink] New post   24 Sep 2014, 03:30
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gmatquant25 wrote:
How many factors of 240 are also multiples of 3 ?

A. 5
B. 8
C. 9
D. 10
E. 20


Find the number of all the factors of 240 and subtract the factors which are multiples of 3 to get the number of factors which are not multiples of 3.

Factorize 240: \(240=2^4*3*5\). It has (4 + 1)(1 + 1)(1 + 1) = 20 factors.

Get rid of 3, we get \(80 =2^4*5\). It has (4 + 1)(1 + 1) = 10 factors.

Now, all these factors of 80 are factors 240 but are NOT multiples of 3. So, the number of factors of 240 which are also multiples of 3 is 20 - 10 = 10.

Answer: D.
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PareshGmat
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How many factors of 240 are also multiples of 3 ? [#permalink] New post   23 Sep 2014, 19:54
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gmatquant25 wrote:
How many factors of 240 are also multiples of 3 ?

A. 5
B. 8
C. 9
D. 10
E. 20


Answer = D = 10

\(240 = 2^4 * 3^1 * 5^1\)

Factors which are multiples of 3 are as follows

1. 3

2. 3 * 2

3. \(3 * 2^2\)

4. \(3 * 2^3\)

5. \(3 * 2^4\)

6. 3 * 5

7. 3 * 5 * 2

8. \(3 * 5 * 2^2\)

9. \(3 * 5 * 2^3\)

10.\(3 * 5 * 2^4\)
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Re: How many factors of 240 are also multiples of 3 ? [#permalink] New post   23 Sep 2014, 20:03
take factors of 240
240= 2^4*3^1*5^1
Total factors of 240= (4+1)*(1+1)*(1+1)
=5*2*2
=20

out of total factors of 20, half will have 0 as power of 3 and half will have 1 as power of 3. It is because we have considered only 0 and 1 as power of 3 to compute total factors of 240.
Therefore factors that are multiple of 3= 20/2= 10

Ans - D
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chuckkkkk
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Re: How many factors of 240 are also multiples of 3 ? [#permalink] New post   28 May 2016, 13:15
Bunuel wrote:
gmatquant25 wrote:
How many factors of 240 are also multiples of 3 ?

A. 5
B. 8
C. 9
D. 10
E. 20


Find the number of all the factors of 240 and subtract the factors which are multiples of 3 to get the number of factors which are not multiples of 3.

Factorize 240: \(240=2^4*3*5\). It has (4 + 1)(1 + 1)(1 + 1) = 20 factors.

Get rid of 3, we get \(80 =2^4*5\). It has (4 + 1)(1 + 1) = 10 factors.

Now, all these factors of 80 are factors 240 but are NOT multiples of 3. So, the number of factors of 240 which are also multiples of 3 is 20 - 10 = 10.

Answer: D.


-----------
Hi Bunuel, thanks for you post!
But could you please elaborate these steps:
Factorize 240: \(240=2^4*3*5\). It has (4 + 1)(1 + 1)(1 + 1) = 20 factors.

Get rid of 3, we get \(80 =2^4*5\). It has (4 + 1)(1 + 1) = 10 factors.


Thank you.
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Re: How many factors of 240 are also multiples of 3 ? [#permalink] New post   29 May 2016, 02:21
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chuckkkkk wrote:
Bunuel wrote:
gmatquant25 wrote:
How many factors of 240 are also multiples of 3 ?

A. 5
B. 8
C. 9
D. 10
E. 20


Find the number of all the factors of 240 and subtract the factors which are multiples of 3 to get the number of factors which are not multiples of 3.

Factorize 240: \(240=2^4*3*5\). It has (4 + 1)(1 + 1)(1 + 1) = 20 factors.

Get rid of 3, we get \(80 =2^4*5\). It has (4 + 1)(1 + 1) = 10 factors.

Now, all these factors of 80 are factors 240 but are NOT multiples of 3. So, the number of factors of 240 which are also multiples of 3 is 20 - 10 = 10.

Answer: D.


-----------
Hi Bunuel, thanks for you post!
But could you please elaborate these steps:
Factorize 240: \(240=2^4*3*5\). It has (4 + 1)(1 + 1)(1 + 1) = 20 factors.

Get rid of 3, we get \(80 =2^4*5\). It has (4 + 1)(1 + 1) = 10 factors.


Thank you.


Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Check for more here: math-number-theory-88376.html
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Re: How many factors of 240 are also multiples of 3 ? [#permalink] New post   30 May 2016, 04:15
240 = 2^4 *3 * 5
total factors = 5*2*2 =20
for a factor to be multiple of 3 , it should have 3 in the no of prime factors
total factors which are not multiple of 3 = 5*2 =10 (factors of 2^4*5 =80)
therefore factors of 3 = total - factors which are not multiple of 3 = 20 -10 =10
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Re: How many factors of 240 are also multiples of 3 ? [#permalink] New post   09 Apr 2024, 11:49
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Re: How many factors of 240 are also multiples of 3 ? [#permalink]
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