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## Monday, 23 June 2014

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