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# Combinatorics: Factorials- Basic Definitions

If $$m$$ and $$k$$ are positive integers, is $$m!+8k$$ a multiple of $$k$$? >(1) $$k \lt m$$ >(2) $$m=3k$$
Correct. [[snippet]] Stat. (1): since $$k\lt m$$, you know that $$m!$$ includes $$k$$ as a factor. For example, if $$k=2$$ and $$m=3$$, then $$m! = 3\cdot 2\cdot 1$$ is a multiple of $$k$$. Therefore, both $$m!$$ and $$8k$$ are multiples of $$k$$, and their sum _must_ also be a multiple of $$k$$. **Stat.(1) → Yes → S → AD**. Stat. (2): $$m$$ is equal to >$$k \times \text{some integer}$$ and must be a multiple of $$k$$. Therefore, both $$m!$$ and $$8k$$ are multiples of $$k$$, and their sum _must_ also be a multiple of $$k$$. **Stat.(2) → Yes → S → D**.
Incorrect. [[snippet]] Yes, Stat. (1) is __sufficient__. But what about Stat. (2)?
Incorrect. [[snippet]] Yes, Stat. (2) is __sufficient__. But what about Stat. (1)?
Incorrect. [[snippet]] In this question, either one of the statements is __sufficient__.
Incorrect. [[snippet]] In this question, either one of the statements is __sufficient__.
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.