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# Integers: Factoring - Factors Vs. Multiples

If $$m$$ and $$k$$ are non-zero integers, is $$m$$ a multiple of $$k$$? >(1) $$\frac{m^2+m}{k}$$ is an integer. >(2) $$m =2 k^2-3 k$$
Incorrect. [[snippet]] According to Stat. (1), $$\frac{m(m+1)}{k}$$ is an integer. You can use __Plugging In__: * If $$m=3$$ and $$k=6$$, then $$\frac{m(m+1)}{k} = \frac{3(3+1)}{6} = 2$$. However, 3 is _not_ a multiple of 6, which makes the answer "No." * If $$m=4$$ and $$k=2$$, then $$\frac{m(m+1)}{k} = \frac{4(4+1)}{2} = 10$$. In this case, 4 _is_ a multiple of 2, which makes the answer "Yes." Therefore, **Stat.(1) → Maybe → IS → BCE**. Now, Stat. (1) is insufficient, but what about Stat. (2)?
Incorrect. [[snippet]] According to Stat. (2), $$m=2k^2−3k$$. Both terms on the right-hand side are multiples of $$k$$. A multiple of $$k$$ minus a multiple of $$k$$ equals a multiple of $$k$$. Hence, $$m$$ must also be a multiple of $$k$$. Therefore, **Stat.(2) → Yes → S**. Now, Stat. (2) is sufficient, but what about Stat. (1)?
Correct. [[snippet]] According to Stat. (1), $$\frac{m(m+1)}{k}$$ is an integer. You can use __Plugging In__: * If $$m=3$$ and $$k=6$$, then $$\frac{m(m+1)}{k} = \frac{3(3+1)}{6} = 2$$. However, 3 is _not_ a multiple of 6, which makes the answer "No." * If $$m=4$$ and $$k=2$$, then $$\frac{m(m+1)}{k} = \frac{4(4+1)}{2} = 10$$. In this case, 4 _is_ a multiple of 2, which makes the answer "Yes." Therefore, **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$m=2k^2−3k$$. Both terms on the right-hand side are multiples of $$k$$. A multiple of $$k$$ minus a multiple of $$k$$ equals a multiple of $$k$$. Hence, $$m$$ must also be a multiple of $$k$$. Therefore, **Stat.(2) → Yes → S → B**.
Incorrect. [[snippet]] According to Stat. (1), $$\frac{m(m+1)}{k}$$ is an integer. You can use __Plugging In__: * If $$m=3$$ and $$k=6$$, then $$\frac{m(m+1)}{k} = \frac{3(3+1)}{6} = 2$$. However, 3 is _not_ a multiple of 6, which makes the answer "No." * If $$m=4$$ and $$k=2$$, then $$\frac{m(m+1)}{k} = \frac{4(4+1)}{2} = 10$$. In this case, 4 _is_ a multiple of 2, which makes the answer "Yes." Therefore, **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] According to Stat. (2), $$m=2k^2−3k$$. Both terms on the right-hand side are multiples of $$k$$. A multiple of $$k$$ minus a multiple of $$k$$ equals a multiple of $$k$$. Hence, $$m$$ must also be a multiple of $$k$$. Therefore, **Stat.(2) → Yes → S**. Now, Stat. (2) is sufficient, but what about Stat. (1)?
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.