Simple proof regarding integers

[Yosty22]

Homework Statement

Show that if m and n are integers such that 4|m²+n², then 4|mn

Homework Equations

The Attempt at a Solution

Since 4 divides m²+n², then we can say that m²+n² = 4k, where k is an integer. I haven't done any mathematical proofs of any kind yet, but we were supposed to see if we could do this. I am kind of stuck as to where to go from here. The only thing I could think of is to try to introduce a m*n term into the equation m²+n² = 4k. To do this, I multiplied and divided the left-hand side of the equation by mn to get: mn(m²+n²/mn) = 4k. Since I am looking to show that 4|mn, or mn = 4*(some integer), I tried to isolate the mn term. That means 4k/(m²+n²/mn) = mn, or 4kmn/(m²+n²) = mn. However, I don't think this proves anything. Any advice on how to proceed (or even where to start)?

Thanks.

 

[MostlyHarmless]

First, when working with integers, in general, dividing is not a good idea. Since you know nothing about $m^2+n^2$, $m^2+n^2/mn$ isn't necessarily an integer.

First, think about how you need you're proof to end. You want to show that mn can be written as an integer multiple of 4.
After that, where you go next really depends on what you are allowed to assume about integers. Are there any results that may be relevant that you've proven in class?

 

[Yosty22]

There is nothing specifically we talked about in class (We haven't really done any proofs at all - we have just talked about Sentential Logic, truth tables, and basic set theory stuff). Although we haven't really covered proofs in class yet, it is considered a "proofs class" so he wanted us to be able to work this out. I know in the end, I want to find that mn = 4l, where l is another integer. However, I can't see how I can isolate a (m*n) term in m²+n² = 4k (especially without dividing). To step through the proof in a logical way, I would assume you have to start with (m²+n²) = 4k and do things to the equation until you get something of the form m*n = 4l. I just have no idea how I would go about doing it at all.

 

[MostlyHarmless]

Try some concrete examples and see if you can't convince yourself why its true.

 

[MostlyHarmless]

It sounds like your prof doesn't really expect an elegant solution, and he really just wants to what you can discover on your own. I wouldn't want to deprive you of that.

 

[Mark44]

First, when working with integers, in general, dividing is not a good idea. Since you know nothing about $m^2+n^2$, $m^2+n^2/mn$ isn't necessarily an integer.

If you meant $\frac{m^2 + n^2}{mn}$, you need parentheses in what you wrote. Without them, what you wrote is $m^2 + \frac{n^2}{mn}$. Also m²/mn is ambiguous, as it could be interpreted as m²/(mn) or as (m²/m) * n.

First, think about how you need you're proof to end. You want to show that mn can be written as an integer multiple of 4.
After that, where you go next really depends on what you are allowed to assume about integers. Are there any results that may be relevant that you've proven in class?