Integers: Prime Numbers

If $$z$$ is a positive integer smaller than integer $$n$$, is $$z$$ a factor of $$n$$? >(1) $$n$$ is divisible by all the positive integers less than or equal to 9. >(2) $$z$$ is not divisible by any prime number.
Incorrect. [[snippet]] Stat. (2) is devious, but what it really means is that $$z=1$$. The only positive number that isn't divisible by any prime number is 1. In order to understand this concept, first recall that any positive integer greater than 1 is either prime (option A) or can be broken down into prime numbers (option B): * Option A: A prime number is always divisible by itself (for example, 7 is divisible by 7). So if $$z$$ is a prime number, it is—by definition—divisible by a prime number, which contradicts the statement. Hence, $$z$$ cannot be prime. * Option B: Any nonprime integer greater than 1 can be broken down into prime numbers. For example, 15 can be broken down into $$3 \times 5$$ and is therefore divisible by 5 and 3. Hence, $$z$$ cannot be a nonprime integer greater than 1 or else it would be divisible by a prime number, contradicting the statement. Thus, the only positive value for $$z$$ (that is not divisible by any prime number) is 1. Since 1 is a factor of any integer (any number is divisible by 1), $$z$$ _must_ be a factor of $$n$$, and the answer to the question is a definite "__Yes__." Therefore **Stat.(2) → Yes → S → BD**.
Correct. [[snippet]] According to Stat. (1), $$n$$ contains all the factors from 1 through 9, inclusive. __Plug In__ for $$z$$. If $$z=5$$, then $$z$$ is indeed a factor of $$n$$ (since $$n$$ must be divisible by 5), and the answer is "__Yes__." However, is the answer always "__Yes__"? The variable $$z$$ could also be (for example) a prime number greater than 9, in which case it isn't a factor of $$n$$. For example, if $$n=9!$$ (which satisfies the question stem) and $$z=13$$, then $$z$$ is _not_ a factor of $$n$$, and the answer is "__No__." Therefore **Stat.(1) → Maybe → IS → BCE**. Stat. (2) is devious, but what it really means is that $$z=1$$. The only positive number that isn't divisible by any prime number is 1. In order to understand this concept, first recall that any positive integer greater than 1 is either prime (option A) or can be broken down into prime numbers (option B): * Option A: A prime number is always divisible by itself (for example, 7 is divisible by 7). So if $$z$$ is a prime number, it is—by definition—divisible by a prime number, which contradicts the statement. Hence, $$z$$ cannot be prime. * Option B: Any nonprime integer greater than 1 can be broken down into prime numbers. For example, 15 can be broken down into $$3 \times 5$$ and is therefore divisible by 5 and 3. Hence, $$z$$ cannot be a nonprime integer greater than 1 or else it would be divisible by a prime number, contradicting the statement. Thus, the only positive value for $$z$$ (that is not divisible by any prime number) is 1. Since 1 is a factor of any integer (any number is divisible by 1), $$z$$ _must_ be a factor of $$n$$, and the answer to the question is a definite "__Yes__." Therefore **Stat.(2) → Yes → S → B**.
Incorrect. [[snippet]] According to Stat. (1), $$n$$ contains all the factors from 1 through 9, inclusive. __Plug In__ for $$z$$. If $$z=5$$, then $$z$$ is indeed a factor of $$n$$ (since $$n$$ must be divisible by 5), and the answer is "__Yes__." However, is the answer always "__Yes__"? The variable $$z$$ could also be (for example) a prime number greater than 9, in which case it isn't a factor of $$n$$. For example, if $$n=9!$$ (which satisfies the question stem) and $$z=13$$, then $$z$$ is _not_ a factor of $$n$$, and the answer is "__No__." Therefore **Stat.(1) → Maybe → IS → BCE**.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
EACH statement ALONE is sufficient to answer the question asked.
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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