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# Integers: Prime Factors are Building Blocks

If \$\$z\$\$ is a positive integer, is \$\$z=7\$\$? >(1) \$\$z\$\$ is not a factor of \$\$120\$\$. >(2) The largest prime factor of \$\$z\$\$ is \$\$7\$\$.
Correct. [[snippet]] In Stat. (1), if \$\$z\$\$ is not a factor of 120, then 120 is not divisible by \$\$z\$\$. Since \$\$z=7\$\$ definitely satisfies this condition, the answer could be "__Yes__." But does it have to be "Yes"? There are many other numbers that are not factors of 120. For example, 120 is not divisible by 9, so \$\$z\$\$ could also be 9, yielding an answer of "__No__." That's a "__Maybe__," so **Stat.(1) → Maybe → IS → BCE**. In Stat. (2), if \$\$z=7\$\$, then \$\$z\$\$ has only one prime factor—itself. Therefore, \$\$z=7\$\$ satisfies the condition in Stat. (2), so \$\$z\$\$ _could_ equal 7, yielding an answer of "__Yes__." But does it have to be "Yes"? The number 7 is also the largest prime factor of 14 (use the factor tree to find 14's prime factors: 2 and 7). So \$\$z=14\$\$ also satisfies Stat. (2), but then \$\$z\ne 7\$\$, and the answer is "__No__." That's a "__Maybe__," so **Stat.(2) → Maybe → IS → CE**. For Stat. (1+2), try to __Plug In__ the same values you've used for the individual statements. Clearly, \$\$z=7\$\$ satisfies both, so the answer could be "__Yes__." However, \$\$z=14\$\$ also satisfies both statements: 120 is not divisible by 14, and 7 is the greatest prime factor of 14. If \$\$z=14\$\$, it does not equal 7 and the answer is "__No__." It is still a "__Maybe__," so **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] In Stat. (1), if \$\$z\$\$ is not a factor of 120, then 120 is not divisible by \$\$z\$\$. Since \$\$z=7\$\$ definitely satisfies this condition, the answer could be "__Yes__." But does it have to be "Yes"? There are many other numbers that are not factors of 120. For example, 120 is not divisible by 9, so \$\$z\$\$ could also be 9, yielding an answer of "__No__." That's a "__Maybe__," so **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] In Stat. (1), if \$\$z\$\$ is not a factor of 120, then 120 is not divisible by \$\$z\$\$. Since \$\$z=7\$\$ definitely satisfies this condition, the answer could be "__Yes__." But does it have to be "Yes"? There are many other numbers that are not factors of 120. For example, 120 is not divisible by 9, so \$\$z\$\$ could also be 9, yielding an answer of "__No__." That's a "__Maybe__," so **Stat.(1) → Maybe → IS → BCE**. In Stat. (2), if \$\$z=7\$\$, then \$\$z\$\$ has only one prime factor—itself. Therefore, \$\$z=7\$\$ satisfies the condition in Stat. (2), so \$\$z\$\$ _could_ equal 7, yielding an answer of "__Yes__." But does it have to be "Yes"? The number 7 is also the largest prime factor of 14 (use the factor tree to find 14's prime factors: 2 and 7). So \$\$z=14\$\$ also satisfies Stat. (2), but then \$\$z\ne 7\$\$, and the answer is "__No__." That's a "__Maybe__," so **Stat.(2) → Maybe → IS → CE**.
Incorrect. [[snippet]] In Stat. (1), if \$\$z\$\$ is not a factor of 120, then 120 is not divisible by \$\$z\$\$. Since \$\$z=7\$\$ definitely satisfies this condition, the answer could be "__Yes__." But does it have to be "Yes"? There are many other numbers that are not factors of 120. For example, 120 is not divisible by 9, so \$\$z\$\$ could also be 9, yielding an answer of "__No__." That's a "__Maybe__," so **Stat.(1) → Maybe → IS → BCE**. In Stat. (2), if \$\$z=7\$\$, then \$\$z\$\$ has only one prime factor—itself. Therefore, \$\$z=7\$\$ satisfies the condition in Stat. (2), so \$\$z\$\$ _could_ equal 7, yielding an answer of "__Yes__." But does it have to be "Yes"? The number 7 is also the largest prime factor of 14 (use the factor tree to find 14's prime factors: 2 and 7). So \$\$z=14\$\$ also satisfies Stat. (2), but then \$\$z\ne 7\$\$, and the answer is "__No__." That's a "__Maybe__," so **Stat.(2) → Maybe → IS → CE**. For Stat. (1+2), try to __Plug In__ the same values you've used for the individual statements. Clearly, \$\$z=7\$\$ satisfies both, so the answer could be "__Yes__." However, \$\$z=14\$\$ also satisfies both statements: 120 is not divisible by 14, and 7 is the greatest prime factor of 14. If \$\$z=14\$\$, it does not equal 7 and the answer is "__No__." It is still a "__Maybe__," so **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] In Stat. (2), if \$\$z=7\$\$, then \$\$z\$\$ has only one prime factor—itself. Therefore, \$\$z=7\$\$ satisfies the condition in Stat. (2), so \$\$z\$\$ _could_ equal 7, yielding an answer of "__Yes__." But does it have to be "Yes"? The number 7 is also the largest prime factor of 14 (use the factor tree to find 14's prime factors: 2 and 7). So \$\$z=14\$\$ also satisfies Stat. (2), but then \$\$z\ne 7\$\$, and the answer is "__No__." That's a "__Maybe__," so **Stat.(2) → Maybe → IS → ACE**.
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.