This week, we continued discussing infinite sets of numbers, bijections, and modular arithmetic (alongside Pythagorean triples).

Here are some challenge problems to consider!

1. Are there natural numbers $a, b$ such that $\frac{a}{b} = \sqrt{2}$?  If so, find them, if not, explain why.
2. Are there $a, b, c, n$ such that $a^2+b^2=c^2$ and exactly two of a,b,c are divisible by n?  If so, find an example, if not, provide an argument why.
3. Again let $a, b, c$ be a Pythagorean triple.  Show that if $c$ is not divisible by $7$, then neither are $a$ or $b$.
4. How many surjections are there from $\mathbb{Z}_7$ to $\mathbb{Z}_6$?  How about surjections from $\mathbb{Z}_n$ to $\mathbb{Z}_{n+1}$?

### One Response to SMA109: Week Three

1. kirui cheruiyot says: