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# Integers: Prime Numbers

If $$x$$, $$y$$, and $$z$$ are positive integers, is the product of $$x$$, $$y$$, and $$z$$ a prime number? >(1) $$xz$$ is a prime number. >(2) $$yz$$ has only one positive factor.
According to Stat. (2), $$yz$$ has only one factor. There's only one integer that fits this description: $$yz = 1$$, which means that $$y = z = 1$$. That's great, but you still don't know whether the third factor, $$x$$, is prime or not. Plug in for the variables: * If $$x = 2$$, $$y = 1$$, and $$z = 1$$, then $$xyz=2$$, which is prime. * If $$x = 4$$, $$y = 1$$, and $$z = 1$$, then $$xyz=4$$, which is not prime. There is no definite answer, so **Stat.(2) → Maybe → IS → CE**. According to Stat. (1+2), $$xz$$ is prime, and $$y = z = 1$$. So $$x$$ is a prime number, and the other two are equal to 1. Both conditions are satisfied, and the answer to the question stem is a definite "Yes." This is a definite answer, so **Stat.(1+2) → Yes → S → C**.
Incorrect. [[snippet]] According to Stat. (1+2), $$xz$$ is prime, and $$y = z = 1$$. So $$x$$ is a prime number, and the other two are equal to 1. Both conditions are satisfied, and the answer to the question stem is a definite "Yes." This is a definite answer, so **Stat.(1+2) → Yes → S → C**.
Incorrect. [[snippet]] According to Stat. (1), $$xz$$ is a prime number, which means that one of the variables is prime and the other is 1. However, that still doesn't tell you anything about the missing integer $$y$$. Plug in for the variables: * If $$x = 3$$, $$y = 2$$, and $$z = 1$$, then $$xyz=6$$, which is not prime. * If $$x = 3$$, $$y = 1$$, and $$z = 1$$, then $$xyz=3$$, which is prime. There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$yz$$ has only one factor. There's only one integer that fits this description: $$yz = 1$$, which means that $$y = z = 1$$. That's great, but you still don't know whether the third factor, $$x$$, is prime or not. Plug in for the variables: * If $$x = 2$$, $$y = 1$$, and $$z = 1$$, then $$xyz=2$$, which is prime. * If $$x = 4$$, $$y = 1$$, and $$z = 1$$, then $$xyz=4$$, which is not prime. There is no definite answer, so **Stat.(2) → Maybe → IS → CE**.
Incorrect. [[snippet]] According to Stat. (2), $$yz$$ has only one factor. There's only one integer that fits this description: $$yz = 1$$, which means that $$y = z = 1$$. That's great, but you still don't know whether the third factor, $$x$$, is prime or not. Plug in for the variables: * If $$x = 2$$, $$y = 1$$, and $$z = 1$$, then $$xyz=2$$, which is prime. * If $$x = 4$$, $$y = 1$$, and $$z = 1$$, then $$xyz=4$$, which is not prime. There is no definite answer, so **Stat.(2) → Maybe → IS → CE**.
Incorrect. [[snippet]] According to Stat. (1), $$xz$$ is a prime number, which means that one of the variables is prime and the other is 1. However, that still doesn't tell you anything about the missing integer $$y$$. Plug in for the variables: * If $$x = 3$$, $$y = 2$$, and $$z = 1$$, then $$xyz=6$$, which is not prime. * If $$x = 3$$, $$y = 1$$, and $$z = 1$$, then $$xyz=3$$, which is prime. There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Correct. [[snippet]] According to Stat. (1), $$xz$$ is a prime number, which means that one of the variables is prime and the other is 1. However, that still doesn't tell you anything about the missing integer $$y$$. Plug in for the variables: * If $$x = 3$$, $$y = 2$$, and $$z = 1$$, then $$xyz=6$$, which is not prime. * If $$x = 3$$, $$y = 1$$, and $$z = 1$$, then $$xyz=3$$, which is prime. There is no definite answer, so **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.