Distinct Factors

Question

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?

Chethan92 · Answer

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

KarishmaB · Answer

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

Aamirso · Answer

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.

ccooley · Answer

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.

I'll also answer your question about distinct factors below! 
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.

IanStewart · Answer

This is not correct. If a question asks "how many prime factors does 12 have?", the answer is two, not three. "Prime factors" and "distinct prime factors" mean the same thing, both on the GMAT and in math everywhere else. There is no logical reason to count the '2' twice when counting the prime divisors of 12 -- it has two prime divisors, 2 and 3.

In very rare questions, the GMAT does want you to count prime divisors with repetition, but any such question needs to be worded to make it very clear that's what the question wants, since it's not a normal interpretation.

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Distinct Factors

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