<s>[INST] What is the biggest $n$ in $4^n$ that divides $7^{2048} - 1$? A few days ago I stumbled on the following question, it was used in the Museum of mathematics masters tournament:
What is the biggest integer $n$ in $4^n$, that divides $7^{2048} - 1$?
a) 1
b) 3
c) 5
d) 7
It was not phrased exactly like this, but it is similar enough.
Using ghci and a python terminal I guessed the answer is 7.
But I don't know how to go about it using only pen and paper.
 [/INST] You can write
\begin{align*}
7^{2048} - 1 &= 7^{2048} - 1^{2048} \\
&= (7^{1024} + 1)(7^{512} + 1)(7^{256} + 1)(7^{128} + 1)(7^{64} + 1)(7^{32} + 1)(7^{16} + 1)(7^{8} + 1)(7^{4} + 1)(7^{2} + 1)(7 + 1)(7-1)
\end{align*}
For all $k \geq 2$, $7^{2^k} + 1 \equiv (-1)^{2^k} + 1 \equiv 2 \mod 4$, so each term of the form $(7^{2^k} + 1)$ has only one factor of 2. The $(7+1)$ term has 3 factors of 2 and the $(7-1)$ has one factor of 2. So the highest power of 2 that divides $7^{2048} + 1$ is $1\cdot 10 + 3 + 1 =14$, which means for you $n=\lfloor 14/2 \rfloor = 7$.  
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