To find the modular inverse of $9$ modulo $16$, we need to find an integer $x$ such that:

$9x \equiv 1 \pmod{16}$

This means we need to solve the equation $9x = 1 + 16k$ for some integer $k$, or equivalently:

$9x \equiv 1 \pmod{16}$

We can find this by trying different values of $x$ until we find one that satisfies the equation. Let's calculate:

$9 \times 1 = 9 \quad (\text{mod } 16) = 9$ $9 \times 2 = 18 \quad (\text{mod } 16) = 2$ $9 \times 3 = 27 \quad (\text{mod } 16) = 11$ $9 \times 4 = 36 \quad (\text{mod } 16) = 4$ $9 \times 5 = 45 \quad (\text{mod } 16) = 13$ $9 \times 6 = 54 \quad (\text{mod } 16) = 6$ $9 \times 7 = 63 \quad (\text{mod } 16) = 15$ $9 \times 8 = 72 \quad (\text{mod } 16) = 8$ $9 \times 9 = 81 \quad (\text{mod } 16) = 1$

Thus, $9 \times 9 \equiv 1 \pmod{16}$, which means the modular inverse of $9$ modulo $16$ is:

$x = 9$

So, $9^{-1} \equiv 9 \pmod{16}$.

Would you like more details or have any questions?

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Related Questions:

1. How can the Extended Euclidean Algorithm be used to find the modular inverse?
2. What is the modular inverse of 7 modulo 16?
3. How do you solve a system of linear congruences using modular inverses?
4. What are the properties of modular inverses in general?
5. How does modular arithmetic apply in cryptography?

Tip: The modular inverse of a number exists if and only if the number and the modulus are coprime (i.e., their greatest common divisor is 1).