What is the largest prime factor of $2^{12}-1$?

$2^{12}=4096$, so the number is 4095. Divide step by step: $4095=5\times 819,819=3\times 273,273=3\times 91$ and $91=7\times 13$. Hence $4095=3^2\times 5\times 7\times 13$ and the largest prime factor is 13. The trap is 91 — it is the largest factor you meet on the way down, but it is composite, not prime.