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Composite Notions
Age 14 to 16
Challenge Level 
The following solution was recieved from Andrei of School 205 Bucharest. Well done and thank you Andrei.
10201 could be written (in base $x$) as:
$$\begin{align*}10201 &= 1x^0 + 2x^2 + 1x^4 \\ &= x^4 + 2x^2 + 1 \\ &= (x^2 + 1)^2 \end{align*}$$
Now, I write 10101 in a similar manner, in base $y$:
$$\begin{align*} 10101& = y^4 + y^2 + 1\\ & = y^4 + 2y^2 - y^2 + 1\\ & = (y^4 + 2y^2 + 1) - y^2\\ & = (y^2 + 1)^2 - y^2\\ & = (y^2 + 1 + y)(y^2 + 1 -y) \end{align*}$$
Therefore both expressions can be factorised, so they are composite.
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