<s>[INST] Difference of squares Determine if $5^{36} - 1$ is divisible by $13$ using the difference of squares.  I tried splitting it up by difference of squares a couple of times but can't seem to get to a point where I can determine divisibility.
 [/INST] We have $5^{36}-1=(5^{18}+1)(5^{18}-1)$. Also, $5^{18}+1=(5^6)^3+1=(5^6+1)((5^6)^2-5^6+1)$, and $5^6+1=(5^2)^3+1=(5^2+1)((5^2)^2-5^2+1)$. We are done, because $5^2+1=26=2\cdot13$.
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