1 problem found
The integers \(a,b\) and \(c\) satisfy \[ 2a^{2}+b^{2}=5c^{2}. \] By considering the possible values of \(a\pmod5\) and \(b\pmod5\), show that \(a\) and \(b\) must both be divisible by \(5\). By considering how many times \(a,b\) and \(c\) can be divided by \(5\), show that the only solution is \(a=b=c=0.\)
Solution: \begin{array}{c|ccccc} a & 0 & 1 & 2 & 3 & 4 \\ a^2 & 0 & 1 & 4 & 4 & 1 \end{array} Therefore \(a^2 \in \{0,1,4\}\) and so we can have \begin{array} $2a^2+b^2 & 0 & 1 & 4 \\ \hline 0 & 0 & 1 & 4 \\ 1 & 2 & 3 & 1 \\ 4 & 3 & 4 & 2 \end{array} Therefore the only solution must have \(5 \mid a,b\), but then we can write them has \(5a'\) and \(5b'\) so the equation becomes \(2\cdot25 a'^2 + 25b'^2 = 5c^2\) ie \(5 \mid c^2 \Rightarrow 5 \mid c\). But that means we can always divide \((a,b,c)\) by \(5\), which is clearly a contradiction if we consider the lowest power of \(5\) dividing \(a,b,c\) for any solution.