To find the multiplicative inverse of $a \mod p$, we need to find an integer $x$ such that:

$a \cdot x \equiv 1 \pmod{p}$

In this case:

• $a = 5$
• $p = 17$

We are looking for $x$ such that:

$5 \cdot x \equiv 1 \pmod{17}$

Method: Using the Extended Euclidean Algorithm

1. Apply the Euclidean algorithm to find the greatest common divisor (GCD) of $a$ and $p$. Since $a = 5$ and $p = 17$ are coprime (GCD is 1), the extended Euclidean algorithm will help us express 1 as a linear combination of 5 and 17.

We perform the Euclidean algorithm steps:

• $17 = 3 \times 5 + 2$
• $5 = 2 \times 2 + 1$
• $2 = 2 \times 1 + 0$

Now, reverse the steps to express 1 as a linear combination of 5 and 17:

• From $1 = 5 - 2 \times 2$, and $2 = 17 - 3 \times 5$, substitute:
• $1 = 5 - 2 \times (17 - 3 \times 5)$
• Simplify: $1 = 5 - 2 \times 17 + 6 \times 5$
• Combine like terms: $1 = 7 \times 5 - 2 \times 17$

Thus, $1 = 7 \times 5 - 2 \times 17$, so $x = 7$.

Conclusion:

The multiplicative inverse of $5 \mod 17$ is $7$.

You can verify this by checking: $5 \times 7 = 35 \quad \text{and} \quad 35 \mod 17 = 1$

Would you like more details or have any questions?

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Here are 5 related questions:

1. How is the extended Euclidean algorithm used to find the greatest common divisor (GCD)?
2. What is the role of prime numbers in cryptographic key generation?
3. Can the modular inverse always be found for any number $a$ modulo $p$ where $p$ is prime?
4. What are the properties of modular arithmetic that make it useful for cryptography?
5. How does modular arithmetic work with non-prime moduli?

Tip: In cryptography, the modular inverse is crucial for algorithms like RSA, where private keys are computed using inverse operations.