hazelnut01 wrote:If k is a positive Integer, how many unique Prime Factors Does 14k have ?
(1) k^4 is divisible by 100
(2) 50*k has 2 Prime Factors
OA=C
First, REPHRASE the question:
We know how many unique prime factors 14 has: two (2 and 7). It would not be enough to know how many unique prime factors k has, since k may also contain factors of 2 or 7. These wouldn't be unique.
Target question: how many prime factors other than 2 or 7 does k have?
(1) k^4 is divisible by 100
Knowing one thing that a number is divisible by gives us only partial information. We know that it contains at least a 2 and a 5, but we don't know what else it may or may not be divisible by.
(2) 50*k has 2 Prime Factors
As GMATinsight points out, this tell us that k has no factors *other than* 2 and 5, but it could contain both, only one of the two, or neither (if k = 1). This doesn't give us enough to answer the question:
- if k contains 2 and 5, 14k will have 3 unique factors: 2,5, 7
- if k contains a 5 only, 14k will have 3 unique factors: 2,5, 7
- if k contains a 2 only, 14k will have 2 unique factors: 2,7
- if k contains neither (k = 1), 14k will have 2 unique factors: 2,7
Together:
k must contain a 2 and a 5 (per stmt 1), but if does not contain any other factors (per stmt 2). Therefore, 14k must contain unique factors of only 2, 5, and 7. Sufficient.
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