3. Compute the greatest common divisor of $4^{8}-1$ and $8^{12}-1$.
Answer: 15 Let $d=\operatorname{gcd}(a, b)$ for some $a, b \in \mathbb{Z}^{+}$. Then, we can write $d=a x-b y$, where $x, y \in \mathbb{Z}^{+}$, and
$$
\begin{array}{l}
2^{a}-1 \mid 2^{a x}-1 \\
2^{b}-1 \mid 2^{b y}-1
\end{array}
$$

Multiplying the right-hand side of (2) by $2^{d}$, we get,
$$
2^{b}-1 \mid 2^{a x}-2^{d}
$$

Thus, $\operatorname{gcd}\left(2^{a}-1,2^{b}-1\right)=2^{d}-1=2^{\operatorname{gcd}(a, b)}-1$.

Using $a=16$ and $b=36$, we get
$$
\operatorname{gcd}\left(2^{16}-1,2^{36}-1\right)=2^{\operatorname{gcd}(16,36)}-1=2^{4}-1=15
$$