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# Distinct Factors

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Manager
Joined: 17 Jul 2017
Posts: 193

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28 Oct 2018, 02:43
Can anyone help me understand the difference between factors and distinct factors?

https://gmatclub.com/forum/if-k-is-a-po ... 58398.html

in this question,
i proceeded like:

20k=2^2*5*k
since k is prime it will always have 2 factors
so answer must be
(2+1)(1+1)(1+1)
last 1+1 is for 2 factors of k
Where is my reasoning wrong?

What is the differnece between factors of a number and distinct factors ?

12 has 2*2*3=2^2*3
i.e (2+1)*(1+1)=6 factors
Are these distinct ?
1,2,3,4,6,12 are all distinct only.So what are non distinct factors?
NUS School Moderator
Joined: 18 Jul 2018
Posts: 1021
Location: India
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)
Re: Distinct Factors  [#permalink]

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28 Oct 2018, 03:10
vanam52923 wrote:
Can anyone help me understand the difference between factors and distinct factors?

https://gmatclub.com/forum/if-k-is-a-po ... 58398.html

in this question,
i proceeded like:

20k=2^2*5*k
since k is prime it will always have 2 factors
so answer must be
(2+1)(1+1)(1+1)
last 1+1 is for 2 factors of k
Where is my reasoning wrong?

What is the differnece between factors of a number and distinct factors ?

12 has 2*2*3=2^2*3
i.e (2+1)*(1+1)=6 factors
Are these distinct ?
1,2,3,4,6,12 are all distinct only.So what are non distinct factors?

Hi,

For the first question. k is a prime. 20 = $$2^2$$$$5^1$$. If k = 2 then 20K = $$2^3$$$$5^1$$ = 4*2 = 8. or k be 5 then no. of factors would be 8. for other primes it'll be 3*2*2 = 12.

For distinct factors. consider a number 12. factors are (2,2,3) only 2 and 3 are distinct here.

Hope it helps
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Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9706
Location: Pune, India
Re: Distinct Factors  [#permalink]

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29 Oct 2018, 02:06
vanam52923 wrote:
Can anyone help me understand the difference between factors and distinct factors?

https://gmatclub.com/forum/if-k-is-a-po ... 58398.html

in this question,
i proceeded like:

20k=2^2*5*k
since k is prime it will always have 2 factors
so answer must be
(2+1)(1+1)(1+1)
last 1+1 is for 2 factors of k
Where is my reasoning wrong?

What is the differnece between factors of a number and distinct factors ?

12 has 2*2*3=2^2*3
i.e (2+1)*(1+1)=6 factors
Are these distinct ?
1,2,3,4,6,12 are all distinct only.So what are non distinct factors?

I have explained the problem with the first question on the link. k must be a prime number so it could be 2 or 5 too. In that case, the number of factors is different.

Total number of factors of the number are the distinct factors as you showed here:
12 has 2*2*3=2^2*3
i.e (2+1)*(1+1)=6 factors
Are these distinct ?
1,2,3,4,6,12 are all distinct only.

The concept of distinct/non distinct rarely comes into play and if it does, it does only when we are talking about prime factors.
12 = 2^2 * 3
Some may say that 12 has 3 prime factors: 2, 2 and 3
But usually when the question asks for number of prime factors, it means the number of distinct prime factors. Just for clarity, the question may mention "what are the number of distinct prime factors of 12?" There are two: 2 and 5
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Karishma
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Intern
Joined: 10 May 2018
Posts: 34
Re: Distinct Factors  [#permalink]

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30 Oct 2018, 13:33
vanam52923 wrote:
Can anyone help me understand the difference between factors and distinct factors?

https://gmatclub.com/forum/if-k-is-a-po ... 58398.html

in this question,
i proceeded like:

20k=2^2*5*k
since k is prime it will always have 2 factors
so answer must be
(2+1)(1+1)(1+1)
last 1+1 is for 2 factors of k
Where is my reasoning wrong?

What is the differnece between factors of a number and distinct factors ?

12 has 2*2*3=2^2*3
i.e (2+1)*(1+1)=6 factors
Are these distinct ?
1,2,3,4,6,12 are all distinct only.So what are non distinct factors?

Regarding your query of factors versus distinct factors, as far as I know, it is totally fine to assume, every factor as distinct factor whenever a question asks you the number of factors. Of course, repeating the factors will lead to awkward, non unique answers.

So whether the question says find number of factors of 20 or number of unique factors of 20, answer shall remain 6.
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 832
GMAT 1: 790 Q51 V49
GRE 1: Q170 V170
Re: Distinct Factors  [#permalink]

### Show Tags

05 Nov 2018, 13:51
vanam52923 wrote:
Can anyone help me understand the difference between factors and distinct factors?

https://gmatclub.com/forum/if-k-is-a-po ... 58398.html

in this question,
i proceeded like:

20k=2^2*5*k
since k is prime it will always have 2 factors
so answer must be
(2+1)(1+1)(1+1)
last 1+1 is for 2 factors of k
Where is my reasoning wrong?

What is the differnece between factors of a number and distinct factors ?

12 has 2*2*3=2^2*3
i.e (2+1)*(1+1)=6 factors
Are these distinct ?
1,2,3,4,6,12 are all distinct only.So what are non distinct factors?

First, you got the problem wrong for a reason that has nothing to do with distinct factors!

Here's why you missed it. The formula you're using is completely correct. In fact, if k = 7 or if k = 11, your reasoning is 100% right. (2+1)(1+1)(1+1) = 12 factors, and that's the number of factors in 20(7) = 140. It's also the number of factors of 20(11) = 220.

The problem is, k could equal 2 or 5. If that's true, your formula doesn't work. If k = 2, then you'd actually be finding the number of factors of (2^3 * 5), which is a different formula: (3+1)(1+1) = 8 factors.

Or if k = 5, you'd be finding the factors of (2^2 * 5^2), which is (2+1)(2+1) = 9 factors.

So, the number of factors isn't always 12. It's only 12 if k is a different prime.

There are actually three different terms, not just two:

1. prime factors (or 'prime factorization')
2. distinct prime factors
3. factors

The prime factors of a number are the primes you get when you 'break down' that number as much as possible. For instance:

12 = 2 * 2 * 3

Prime factors are 2, 2, and 3.

The distinct prime factors of a number are just the unique prime factors, without any repeats.

The distinct prime factors of 12 are 2 and 3.

The factors of a number don't have to be prime at all! If a problem just says 'factors', it's talking about all of the positive integers that can be divided evenly into that number. Not just the primes!

The factors of 12 are 1, 2, 3, 4, 6, and 12.

If a problem says 'different factors', this is what the problem means. It doesn't technically have to say 'different factors'; if it just says 'factors', you should usually assume that it's looking for all of the factors of the number, like 1, 2, 3, 4, 6, 12, and that you shouldn't skip any or count any twice.
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Chelsey Cooley | Manhattan Prep | Seattle and Online

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Re: Distinct Factors   [#permalink] 05 Nov 2018, 13:51
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# Distinct Factors

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