Good solutions were submitted by Hyeyoun Chung, age 15, St Paul's
Girls' School, London, Michal Szymanski age 16, Daramalan College,
Ling Xiang Ning, Allan, age 13, Raffles Institution, Singapore. 
<p>This is Hyeyoun's solution.</p>
<p><!--#perl sub="HTML::TeX" arg="
We are given that \$\sqrt(a)\$  is a natural number and  is rational.
Therefore , where \$x\$ and \$y\$ are integers and \$y \ne 0\$. Then
.
\[
 a = \frac{x^2}{y^2}
\]
\par
We can always write the rational number so that x and y are coprime, that is they have no common factors except 1. As a is a natural number, \$y^2\$ must be a factor of \$x^2\$. Therefore, y must be a factor of x and therefore y=1
\par
Therefore \$a\$ is a square number.
\par
It follows that if a is not a square number then \$\sqrt(a)\$  is irrational.
" --></p>