Explaining maths

See all posts in this series I have mentioned modular arithmetic (sometimes known as "clock arithmetic") in several of my posts. But this may not be helpful if you haven't come across this type of arithmetic before.

In a recent FPM class we discussed how to determine $\mathrm{GCD}(21,2^{70}+5)$. [See also my earlier post on Greatest Common Divisors ] The answer here was that $21 \mathrel{|} 2^{70}+5$ and so $\mathrm{GCD}(21,2^{70}+5) = 21\,.$ We discussed various ways to prove this. Some students had shown separately that $3 \mathrel{|} 2^{70}+5$ and $7 \mathrel{|} 2^{70}+5$, from which it follows, e.g. by prime factorization (Fundamental Theorem of Arithmetic, FTA), or by more elementary means, that $21 \mathrel{|} 2^{70}+5$ too. In the Engagement Session we showed, instead, that $2^{70} \equiv 2^4 = 16$  $(\mathrm{mod}~ 21)$, which gives us $2^{70}+5 \equiv 16+5 = 21 \equiv 0$  $(\mathrm{mod}~ 21)$ and so $21 \mathrel{|} 2^{70}+5$. We discovered that $2^{70} \equiv 2^4$  $(\mathrm{mod}~ 21)$ by looking at powers of $2$ mod $21$, and spotting that we had a repeating pattern with period $6$ , so that $2^{70}=2^{4+66} \equiv 2^4$ \((\mathrm{mod}~ 2...

I recently posted the following on this year's FPM Piazza forum. Hi everyone, I saw this a few years ago on QI, and found another version on the web at http://mathandmultimedia.com/2014/12/02/merry-christmas-equation/ If we are told that $y=\frac{\ln(\frac{x}m-sa)}{r^2}$