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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**.