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10 Jun 2010, 01:33
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
57% (01:35) correct
43%
(01:34)
wrong
based on 1062
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If K is a positive integer, how many different prime numbers are factors of the expression \(K^2\)?
1) Three different prime numbers are factors of \(4K^4\).
2) Three different prime numbers are factors of 4K.
1) Three different prime numbers are factors of \(4K^4\).
2) Three different prime numbers are factors of 4K.
10 Jun 2010, 04:40
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ykaiim
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.
21 Jan 2013, 22:40
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ykaiim
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
