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 \(x\):

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