Exercise (4.3).7
Prove that 3 is a factor of $2^{2n}− 1$.
Using Proposition (4.9) we have
$$ 2 \mid 2n \implies (2^{2}-1) \mid (2^{2n}-1) $$
That is, $2^2-1 = 3$ is a factor of $2^{2n}-1$.
Exercise (4.3).8
Prove that 7 is a factor of $2^{3n}-1$.
Using Proposition (4.9) we have
$$ 3 \mid 3n \implies (2^3-1) \mid (2^{3n}-1) $$
That is, $2^3-1=7$ is a factor of $2^{3n}-1$.
Exercise (4.3).9
Prove that 2047 is a factor of $2^{3751}− 1$.
Using Proposition (4.9) we have
$$ 11 \mid 3751 \implies (2^{11}-1) \mid (2^{3751}-1) $$
That is, $2^{11}-1=2047$ is a factor of $2^{3751}-1$.