<s>[INST] Checking whether the number is composite Prove that $5^{125}-1$/ ($5^{25} - 1$) is composite
I have written $5^{125}-1$ as $(5^{25}-1)(5^{100}+5^{75}+5^{50}+5^{25}+1)$ but what should I do after this? 
Sorry about earlier mistake in question ,
 [/INST] Let $x = 5^{25}$.
$\begin{align}
5^{125}-1 &= x^5-1\\
&=(x^4 +x^3 +x^2 + x + 1)(x-1) \\
&= (x^4 + 9x^2 + 1 + 6x^3 + 6x + 2x^2 - 5x^3 - 10x^2 - 5x)(x - 1)\\
&= ((x^2 + 3x + 1)^2 - 5x(x + 1)^2)(x - 1)
\end{align}$
Put $x = 5^{25}$, just in the expression $5x$, you will get 
$5^{125}-1=((x^2 + 3x + 1)^2 - (5^{13}(x + 1))^2)(x - 1)$
it is now of the form $(a^2-b^2)(x-1)$ where $a,b$ are integers and hence $(x^5-1)/(x-1)$ is a composite number. 
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