Mathematical logic and thinking are grounded in a clear understanding of how the truths of various mathematical statements are linked together.

For example, for any number $x$ the expressions $x> 1$ and $x^2> 1$ are both mathematical statements which might be true or might be false. However, we always know that $x^2> 1$ IF $x> 1$, whereas it is not always the case that $x> 1$ IF $x^2> 1$ (consider $x=-2$, for example). Thus:

It is correct to write $\quad\quad x^2> 1$ IF $x> 1$

It is incorrect to write $\quad\quad x> 1$ IF $x^2> 1$

Test out your logical thinking with these statements where n and m are positive integers (full screen version ).

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Are there multiple solutions? If not, how do you know?

How would the logic change if $n$ and $m$ were not necessarily positive or not necessarily integers?

Extension: Note that this activity does not prove that the statements are true. How might you go about proving that certain combinations are correct? How might you go about proving that certain combinations are incorrect?