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 \(x\):

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