# Composite Notions

A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.

## Problem

Image

Show that 10201 is composite in any base.

Likewise show that 10101 is composite in any base.

**Further Reading:** Learn About Number Bases by Toni
Beardon

## Getting Started

The number 2356 in base 10 can be written

$ 2 \times 10^3 + 3 \times 10^2 + 5 \times 10^1 + 6 \times 10^0 = 2000 + 300 + 50 + 6$

So the number 234561 in base y can be written $2 \times y^5 + 3 \times y^4 + 4\times y^3 + 5 \times y^2 + 6 \times y^1 + 1 \times y^0$

How about factorising?

## Student Solutions

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.