If K is a positive integer, how many different prime numbers

First of all \(k^x\) (where \(x\) is an integer \(\geq{1}\)) will have as many different prime factors as integer \(k\). Exponentiation doesn't "produce" primes.

Next: \(p^y*k^x\) (where \(p\) is a prime and \(y\) is an integer \(\geq{1}\)) will have as many different prime factors as integer \(k\) if \(k\) already has \(p\) as a factor OR one more factor than \(k\)if \(k\) doesn't have \(p\) as a factor .

So, the question basically is: how many different prime numbers are factors of \(k\)?

(1) Three different prime numbers are factors of \(4k^4\) --> if \(k\) itself has 2 as a factor (eg 30) than it's total # of primes is 3 but if k doesn't have 2 as a factor (eg 15) than it's total # of primes is 2. Not sufficient.

(2) Three different prime numbers are factors of \(4k\) --> the same as above: if \(k\) itself has 2 as a factor (eg 30) than it's total # of primes is 3 but if k doesn't have 2 as a factor (eg 15) than it's total # of primes is 2. Not sufficient.

(1)+(2) Nothing new, k can be 30 (or any other number with 3 different primes, out of which one factor is 2) than the answer is 3 or k can be 15 (or any other number with 2 different primes, out of which no factor is 2) than the answer is 2. Not sufficient.

Answer: E.
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I notice that stopping to analyze the question first before going straight to the statements make it easier to solve this DS questions.
The question is looking for the number of distinct prime numbers of K. Whether it be K^4, K^100 or K^9, the number of distinct prime numbers would be the same with just K.

Statement 1: 3 prime number factors for 4K^4. Well there are two possibilites:
(a) either K has "2" and 2 other prime numbers (e.g. 2, 3 and 7; 2, 5 and 11) OR
(b) K just have two prime numbers and no "2" (e.g. 5 and 7)

Thus, we know K could have 2 or 3 distinch prime number factors. INSUFFICIENT.

Statement 2: Once you get used to it, you will notice Statement (2) is just the same as Statement (1).. It throws in "2" outside the K making it blurry whether "2" is within K or not. Thus, INSUFFICIENT.

If Statement (1) and Statement (2) are just similar givens. Then together, they are INSUFFICIENT.

Answer: E

Dang, Bunnuel. I'm about to just study your explanations for the quant section of the GMAT. lol What was your undergrad degree in? GMAT math? I'm jk. Thanks for your help
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K>0
K will have the same number of prime factors as \(K^2\)
So the question is asking for the number of prime factors of K.

1) 4\(K^4\) = \(P^a\)*\(Q^b\)*\(R^c\)
If K = 15, then 4\(K^4\) has 3 factors i.e. 2,3,5 and K has 2 factors i.e. 3 and 5.
If K = 30, then 4\(K^4\) has 3 factors i.e. 2,3,5 BUT K has 3 factors i.e. 2,3 and 5.
Insufficient.

2) 4K= \(P^a\)*\(Q^b\)*\(R^c\)
Same values of K can be used as above. Insufficient.

1+2) Both statements provide the same information. Insufficient.

Answer is E.
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Combining is insufficient:

If k = 4 then there is only one prime number in 4K or 4K^2
If k = 5 then there are two prime numbers in 4K or 4K^2

Another method that worked for me was to find integer K-if we find K we can find its factors.
Both statements have 4 as a multiplier which is not divisible by any of the prime factors on the LHS. Hence K cannot be found as an integer so E. Bunuel, does this logic make sense? Please enlighten!

Tricky question. But it is E as there is no way to know whether 2 was a factor in K or not. Neither solution alone gives a direct hint to this, and together it can't be solved either.

We need to determine the number of different prime factors in K^2. We must remember that the number of distinct prime factors of K^n for any positive integer n is same as the number of distinct prime factors of K. Thus, K^2 has the same distinct prime factors as K. That is, if we know the number of distinct prime factors K has, then we know the number of distinct prime factors K^2 has.

Statement One Alone:

Three different prime numbers are factors of 4K^4.

The information in statement one is not sufficient.

For instance, if K = 2 x 3 x 5, then 4K^4 has 3 different prime factors and K^2 also has 3 different prime factors (since K has 3 different prime factors). However, if K = 3 x 5, then 4K^2 has 3 different prime factors; however, K^2 has 2 different prime factors (since K has 2 different prime factors).

Statement Two Alone:

Three different prime numbers are factors of 4K.

The information in statement two is not sufficient.

For instance, if K = 2 x 3 x 5, then 4K has 3 different prime factors and K^2 has 3 different prime factors. However, if K = 3 x 5, then 4K has 3 different prime factors; however, K has 2 different prime factors.

Statements One and Two Together:

We see that using both statements still allows for K to have 2 or 3 different prime factors.

Answer: E
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