Let $x=5^{25}$, then \begin{align} &\frac{5^{125}-1}{5^{25}-1}=\frac{x^5-1}{x-1}\\ &=x^4+x^3+x^2+x+1\\ &=(x^2 +3x+1)^2 - (5x)(x+1)^2 \end{align} The last expression can be factorized by the difference of square formula as long as $5x$ is a square. In our case, $5x = 5\times 5^{25}=(5^{13})^2$. Therefore, we conclude the given expression can be factorized which means it is composite.