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# Integers: Factoring - Factors Vs. Multiples

If $$x$$ and $$y$$ are non-negative integers, is 11 a factor of $$x + y$$? >(1) 22 is a factor of $$x$$. >(2) $$x - y$$ is divisible by 11.
According to Stat. (2), $$(x - y)$$ is divisible by 11. Does that mean that $$x+y$$ is divisible by 11 as well? That depends on whether $$x$$ and $$y$$ are both multiples of 11 or not. __Plug In__ values for the variables: * If $$x = 22$$ and $$y = 11$$, then 11 _is_ a factor of $$x + y$$, so the answer is "Yes." * If $$x = 13$$ and $$y = 2$$, then $$x-y=11$$, which is divisible by 11. In this case, 11 is _not_ a factor of $$x + y$$, so the answer is "No." There is no definite answer, so **Stat.(2) →Maybe → IS → CE**. According to Stat. (1+2), $$x$$ is divisible by 22 and $$x - y$$ is divisible by 11. According to the rules of adding and subtracting multiples, >$$\text{Multiple of 11} - \text{Multiple of 11} = \text{Multiple of 11}$$. The opposite is also true: if $$x$$ is a multiple of 11 and the result when subtracting $$y$$ is also a multiple, then $$y$$ itself must be a multiple of 11 so as to reach the next multiple from $$x$$. For example, __Plug In__ $$x = 44$$ and $$x-y = 22$$. In this case, $$y$$ must equal 22, another multiple of 11, so as not to "miss" the 22 when subtracting from 44. Once you've proven that both $$x$$ and $$y$$ are multiples of 11, then $$x+y$$ must also be a multiple of 11, and the answer is a definite "Yes," so **Stat.(1+2) → S → C**.
Incorrect. [[snippet]] According to Stat. (1+2), $$x$$ is divisible by 22 and $$x - y$$ is divisible by 11. According to the rules of adding and subtracting multiples, >$$\text{Multiple of 11} - \text{Multiple of 11} = \text{Multiple of 11}$$. The opposite is also true: if $$x$$ is a multiple of 11 and the result when subtracting $$y$$ is also a multiple, then $$y$$ itself must be a multiple of 11 so as to reach the next multiple from $$x$$. For example, __Plug In__ $$x = 44$$ and $$x-y = 22$$. In this case, $$y$$ must equal 22, another multiple of 11, so as not to "miss" the 22 when subtracting from 44. Once you've proven that both $$x$$ and $$y$$ are multiples of 11, then $$x+y$$ must also be a multiple of 11, and the answer is a definite "Yes," so **Stat.(1+2) → S → C**.
Incorrect. [[snippet]] According to Stat. (1), 22 is a factor of $$x$$, so $$x$$ can be 22, 44, 66, and so on. __Plug In__ values for the variables: * If $$x = 22$$ and $$y = 1$$, then 11 is _not_ a factor of $$x + y$$, so the answer is "No." * If $$x = 22$$ and $$y = 11$$, then 11 _is_ a factor of $$x + y$$, so the answer is "Yes." There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$(x - y)$$ is divisible by 11. Does that mean that $$x+y$$ is divisible by 11 as well? That depends on whether $$x$$ and $$y$$ are both multiples of 11 or not. __Plug In__ values for the variables: * If $$x = 22$$ and $$y = 11$$, then 11 _is_ a factor of $$x + y$$, so the answer is "Yes." * If $$x = 13$$ and $$y = 2$$, then $$x-y=11$$, which is divisible by 11. In this case, 11 is _not_ a factor of $$x + y$$, so the answer is "No." There is no definite answer, so **Stat.(2) →Maybe → IS → CE**.
Incorrect. [[snippet]] According to Stat. (1), 22 is a factor of $$x$$, so $$x$$ can be 22, 44, 66, and so on. __Plug In__ values for the variables: * If $$x = 22$$ and $$y = 1$$, then 11 is _not_ a factor of $$x + y$$, so the answer is "No." * If $$x = 22$$ and $$y = 11$$, then 11 _is_ a factor of $$x + y$$, so the answer is "Yes." There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] According to Stat. (2), $$(x - y)$$ is divisible by 11. Does that mean that $$x+y$$ is divisible by 11 as well? That depends on whether $$x$$ and $$y$$ are both multiples of 11 or not. __Plug In__ values for the variables: * If $$x = 22$$ and $$y = 11$$, then 11 _is_ a factor of $$x + y$$, so the answer is "Yes." * If $$x = 13$$ and $$y = 2$$, then $$x-y=11$$, which is divisible by 11. In this case, 11 is _not_ a factor of $$x + y$$, so the answer is "No." There is no definite answer, so **Stat.(2) →Maybe → IS → ACE**.
Correct. [[snippet]] According to Stat. (1), 22 is a factor of $$x$$, so $$x$$ can be 22, 44, 66, and so on. __Plug In__ values for the variables: * If $$x = 22$$ and $$y = 1$$, then 11 is _not_ a factor of $$x + y$$, so the answer is "No." * If $$x = 22$$ and $$y = 11$$, then 11 _is_ a factor of $$x + y$$, so the answer is "Yes." 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.