Sunday Afternoon Maths XVIII Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Parabola

The co-ordinates of the points where the lines intersect the parabola are $(a,a^2)$ and $(-b,b^2)$. Hence the gradient of the line between them is:

$$\frac{a^2-b^2}{a-(-b)}=\frac{(a+b)(a-b)}{a+b}=a-b$$

Therefore the y-coordinate is:

$$b^2 + b(a-b) = ba$$

Ferdinand Möbius, who discovered this property called the curve a Multiplicationsmaschine or 'multipliction machine' as it could be used to perform multiplication.

Extension

How could you use the graph of $y=x^2$ to divide 100 by 7?

Seven Digits

Let's call Dr. Dingo's number $n$. If the number is squared twice then multiplied by $n$, we get $n^5$.

For all integers $n$, the final digit of $n^5$ is the same as the final digit of $n$. In other words:

$$n^5\equiv n \mod 10$$

Therefore, the final digit of Dr. Dingo's number is 7.

$$7^5=16807$$ $$17^5=1419857$$ $$27^5=14348907$$

So, in order for the answer to have seven digits, Dr. Dingo's number was 17.

Extension

For which integers $m$ does there exist an integer $n$ such that for all integers $x$:

$$x^n\equiv x \mod m$$