Solving Divisibility and Remainder Problems

If _m_ is divisible by 2 and _n_ is an integer not divisible by 2, what _must_ be true about _m_ + _n_?

Incorrect. Recall what it means to be divisible by 2. If _m_ is divisible by 2 but _n_ is not, think about what this tells you about _m_ + _n_. Since the question asks what _must_ be true, you can also __Plug In__ numbers to decide which statements are not always true. For example, if _m_ = 4 and _n_ = 3, then $$m+n=7$$ is _not_ divisible by 2.

Alternatively, you could have tried the __Plugging In__ strategy to realize that _m_ + _n_ will always be odd and thus will have a remainder of 1 when divided by 2. For example, if you substitute _m_ with 4 and _n_ with 3, then _m_ + _n_ = 7. Using this value to consider each answer choice, you conclude the following: - The number 7 is not divisible by 2. - When dividing 7 by 2, you get 3 with a remainder of 1 (this is the correct answer choice). - When dividing 7 by 2, the remainder is not 2. - The number 7 is not an even number. - To rule out the last answer choice, you actually needed to consider another example. Keep in mind that since the question is asking for an answer choice that _must_ be true, you only need to find one example that doesn't work with this statement. In this case, _m_ = 0 and _n_ = 3 serves as an example since $$\displaystyle m + n = 0 + 3 = 3 = n$$, which is not greater than _n_.

Incorrect. Think about what it means to get a remainder of 2 when dividing by 2. Try to think of an easier way to say this. You might also want to try and pick appropriate values for _m_ and _n_ and see what must be true about _m_ + _n_. In other words, try using the strategy of __Plugging In__ values. For example, if _m_= 4 and _n_ = 3, then $$m+n=7$$. Now, when dividing 7 by 2, the remainder is 1, not 2.

Incorrect. It is great that you are thinking about divisibility by 2 as being related to even vs. odd numbers. This can really help you in figuring out the answer. You may want to follow this idea and think carefully about whether _m_ and _n_ are even or odd. Once you've decided on that, try using the __Plugging In__ strategy to see whether _m_ + _n_ is even or odd. For example, if _m_ = 4 and _n_ = 3, then $$m+n=7$$ is _not_ even.

Incorrect. Consider the example in which _m_ = 0 and _n_ = 3. In this case, $$\displaystyle m + n = 3 = n$$, which is _not_ greater than _n_.

Great! One way to answer this question is to use the rules for addition of remainders. In order to do this, you need to find the remainders obtained when _m_ and _n_ are divided by 2. You should recall that the remainder when dividing by 2 must be either 0 or 1. Since _m_ is divisible by 2, its remainder will be 0. On the other hand, since _n_ is not divisible by 2, its remainder must be 1. So, using the addition rule for remainders of the same divisor, you know that when _m_ + _n_ is divided by 2, the remainder will be 0 + 1 = 1.

_m + n_ is divisible by 2.

The remainder when dividing _m_ + _n_ by 2 is 1.

The remainder when dividing _m_ + _n_ by 2 is 2.

_m_ + _n_ is even.

_m_ + _n_ is greater than _n_.

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