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How many factors greater than 1 do 120, 210, and 270 have in common?

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How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 04:49
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A
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D
E

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Question Stats:

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Re: How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 04:57
+ IMO option B 3:
How many factors greater than 1 do 120, 210, and 270 have in common?
120: 2^2*3^1*5^1
210: 2^1*3*1^5^2
270:2^1*3^3*5^1

2,3,5 are common factors greater than 1
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How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 16:43
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We are looking for all shared factors of 120, 210, and 270 that are greater than 1.

The greatest common factor of 120, 210, and 270 is 30.

30 = (2)(3)(5)

We need to find all of the factors of the greatest common factor. To do so, we add 1 to each exponent of the prime factors of 30 and then multiply those numbers.

Each prime factor of 30 is only raised to the first power, thus, after increasing each of their powers by 1, the product of their exponents will be:

(2)(2)(2) = 8

There are 8 tots factors of 30. This includes the number 1. We were asked to find all common factors greater than 1. So, we will subtract 1 from 8.

Total number of factors shared among 120, 210, and 270

8 - 1 = 7

To summarize: to find all of the factors shared among 120, 210 and 270, we needed to determine the greatest common factor. Once we determined the GCF to be 30, we needed to find the total number of factors of 30. Lastly, we need to remember that the question asked about the number of factors that are greater than 1.

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How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post Updated on: 13 Aug 2018, 23:25
Bunuel wrote:
How many factors greater than 1 do 120, 210, and 270 have in common?

(A) One
(B) Three
(C) Six
(D) Seven
(E) Thirty


We have GCD(120,210,270)=30

Writing 30 in it's prime factorization form :- \(30=2^1*3^1*5^1\)
No of different factors of 30:- (1+1)*(1+1)*(1+1)=8
These 8 factors is including 1, but we are told to determine the factors greater than 1.

So the required number of factors: 7

Ans. (D)
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Originally posted by PKN on 13 Aug 2018, 22:01.
Last edited by PKN on 13 Aug 2018, 23:25, edited 1 time in total.
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How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 22:17
archish3113 wrote:
+ IMO option B 3:
How many factors greater than 1 do 120, 210, and 270 have in common?
120: 2^2*3^1*5^1
210: 2^1*3*1^5^2
270:2^1*3^3*5^1

2,3,5 are common factors greater than 1


Hey archish3113

Agreed that 2,3, and 5 are the only common prime factors greater than 1.

However, 6(2*3),10(2*5),15(3*5),30(2*3*5) are also common factors which you seem to have forgotten.

Therefore, the total number of common factors are 7(2,3,5,6,10,15, and 30)

Hope this helps you!
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Re: How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 23:01
PKN wrote:
Bunuel wrote:
How many factors greater than 1 do 120, 210, and 270 have in common?

(A) One
(B) Three
(C) Six
(D) Seven
(E) Thirty


We have LCM(120,210,270)=30

Writing 30 in it's prime factorization form :- \(30=2^1*3^1*5^1\)
No of different factors of 30:- (1+1)*(1+1)*(1+1)=8
These 8 factors is including 1, but we are told to determine the factors greater than 1.

So the required number of factors: 7

Ans. (D)


I think you wanted to tell HCF / GCD of 120, 210, 270 is 30. Rather than LCM.

Also, can you please clarify how did you think of taking HCF and then finding its total number of factors rather than prime factorising all numbers and finding the common factors.

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Re: How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 23:24
tatz wrote:
PKN wrote:
Bunuel wrote:
How many factors greater than 1 do 120, 210, and 270 have in common?

(A) One
(B) Three
(C) Six
(D) Seven
(E) Thirty


We have LCM(120,210,270)=30

Writing 30 in it's prime factorization form :- \(30=2^1*3^1*5^1\)
No of different factors of 30:- (1+1)*(1+1)*(1+1)=8
These 8 factors is including 1, but we are told to determine the factors greater than 1.

So the required number of factors: 7

Ans. (D)


I think you wanted to tell HCF / GCD of 120, 210, 270 is 30. Rather than LCM.

Also, can you please clarify how did you think of taking HCF and then finding its total number of factors rather than prime factorising all numbers and finding the common factors.

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Quote:
I think you wanted to tell HCF / GCD of 120, 210, 270 is 30. Rather than LCM.

You are correct, calculated GCD.(NOT LCM)

Quote:
Also, can you please clarify how did you think of taking HCF and then finding its total number of factors rather than prime factorising all numbers and finding the common factors.

Thought process:-
1. We are told to determine the maximum no of common factors of three numbers. (All factors must be greater than 1)
2. If a number is divisible by its highest factor, then that number is also divisible by all of the factors of the highest factor. (
3. What is the magnitude of highest factor of 'n' nos of numbers? GCD is the answer. Then calculate GCD. This is the highest number which divides all the numbers under discussion.
4.Now apply 2 above. Calculate the no of factors of the GCD. The no of factors of a number is calculated using prime factorization method as explained.
5. Since the no of factors is inclusive of 1, so the required no of factors will be 1 less the total no of factors.

Thanking you.
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Re: How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 13 Aug 2018, 23:31
We are looking for all common factors of 120, 210, and 270 that are greater than 1.

The greatest common factor of 120, 210, and 270 is 30.

30=2*3*5

Total of factors formed =2*2*2=8
(Inclusive of 1)

So total no of factors greater than 1 =8-1
7

Option D

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Re: How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 01 Feb 2019, 09:54
Bunuel wrote:
How many factors greater than 1 do 120, 210, and 270 have in common?

(A) One
(B) Three
(C) Six
(D) Seven
(E) Thirty


Breaking each integer into its prime factors, we have:

120 = 12 x 10 = 2^2 x 3 x 5

210 = 21 x 10 = 2 x 3 x 5 x 7

270 = 27 x 10 = 2 x 3^2 x 5

We see that the 3 numbers have the following prime factors in common: 2^1 x 3^1 x 5^1. Adding 1 to each exponent produces (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8 factors, including the number 1. So we have 7 factors that are greater than 1.

Answer: D
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Re: How many factors greater than 1 do 120, 210, and 270 have in common?  [#permalink]

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New post 01 Feb 2019, 10:02
Bunuel wrote:
How many factors greater than 1 do 120, 210, and 270 have in common?

(A) One
(B) Three
(C) Six
(D) Seven
(E) Thirty


120 = 2*2*2*3*5
210 = 2* 3*5*7
270 = 3*3*3*2*5

Now we can just take out the common factors from above
2,3,5,6,10,15,30

Seven

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Re: How many factors greater than 1 do 120, 210, and 270 have in common?   [#permalink] 01 Feb 2019, 10:02
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