Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

DOTS Division

Age 14 to 16Challenge Level

We have received a very clearly explained solution to this, but unfortunately whoever sent it did not include their name. If it was you, let us know!

Let $ab=x$ and $cd=y$ (where $ab$ means $a$ as the tens digit, and $b$ as the ones digit, not $a$ times $b$).

$\frac{abcd^2-cdab^2}{ab^2-cd^2} = \frac{(100x+y)^2 - (100y+x)^2}{x^2-y^2}$ $= \frac{(10000x^2+200xy+y^2) - (10000y^2+200xy+x^2)}{x^2-y^2}$ $= \frac{9999x^2 - 9999y^2}{x^2-y^2}$ $= \frac{9999(x^2-y^2)}{x^2-y^2}$ $= 9999$

(since $x> y$ we are not dividing by zero)