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).

Learn how to find the modular inverse of 9 modulo 16 step-by-step. Understand the concept of modular arithmetic and how to solve equations involving modular inverses. Suitable for high school students studying advanced mathematics.

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