Number theory help please!

Here is the solution to problem 2:

Given that $b$ is the inverse of $a$ modulo 44, we have the relationship:

$$ab \equiv 1 \pmod{44}$$

We need to find the inverse of $9a$ modulo 44 in terms of $b$. Let the inverse of $9a$ modulo 44 be $x$. This means:

$$(9a)x \equiv 1 \pmod{44}$$

We can factor out the $a$:

$$9(ax) \equiv 1 \pmod{44}$$

Since $ab \equiv 1 \pmod{44}$, we replace $ax$ with $b$:

$$9b \equiv 1 \pmod{44}$$

Thus, $x = 9b$ is the inverse of $9a$ modulo 44.

So, the inverse of $9a$ modulo 44 in terms of $b$ is $\boxed{9b}$.