Find the multiplicative inverse of 17 modulo 23. (Demonstrate how you’ve found it. Guessing is not accepted.)

To find the multiplicative inverse of 17 modulo 23, we need to find an integer \( x \) such that:

\[
17x \equiv 1 \pmod{23}
\]

This equation means that \( 17x - 1 \) is divisible by 23, or equivalently:

\[
17x = 1 + 23k
\]

for some integer \( k \). The standard way to find this \( x \) is by using the **Extended Euclidean Algorithm**.

### Step 1: Apply the Euclidean Algorithm
We need to find the greatest common divisor (GCD) of 17 and 23 and express it in the form:

\[
\text{GCD}(17, 23) = 17x + 23y
\]

Since 17 and 23 are coprime (i.e., their GCD is 1), we will use the Extended Euclidean Algorithm to find \( x \).

First, apply the Euclidean algorithm to find the GCD:
\[
23 = 17 \times 1 + 6
\]
\[
17 = 6 \times 2 + 5
\]
\[
6 = 5 \times 1 + 1
\]
\[
5 = 5 \times 1 + 0
\]

So, the GCD is 1 (as expected since 17 and 23 are coprime).

### Step 2: Use the Extended Euclidean Algorithm
Now, we backtrack to express 1 as a linear combination of 17 and 23.

Starting from the second to last equation:
\[
1 = 6 - 5 \times 1
\]

Substitute \( 5 \) from the equation before:
\[
1 = 6 - (17 - 6 \times 2) \times 1
\]
\[
1 = 6 - 17 + 6 \times 2
\]
\[
1 = 6 \times 3 - 17
\]

Now substitute \( 6 \) from the equation \( 6 = 23 - 17 \times 1 \):
\[
1 = (23 - 17 \times 1) \times 3 - 17
\]
\[
1 = 23 \times 3 - 17 \times 3 - 17
\]
\[
1 = 23 \times 3 - 17 \times 4
\]

Thus, we have:
\[
1 = 23 \times 3 - 17 \times 4
\]

### Step 3: Interpret the Result
From this equation, we see that:
\[
-17 \times 4 \equiv 1 \pmod{23}
\]

Multiplying both sides by \(-1\), we get:
\[
17 \times 4 \equiv -1 \equiv 22 \pmod{23}
\]

Thus, the multiplicative inverse of 17 modulo 23 is \( \boxed{19} \), since \( 22 \equiv -1 \pmod{23} \) and \( 23 - 4 = 19 \).

### Verification
We can verify by direct multiplication:
\[
17 \times 19 = 323
\]
Now calculate \( 323 \mod 23 \):
\[
323 \div 23 = 14 \text{ remainder } 1
\]
So,
\[
323 \equiv 1 \pmod{23}
\]

Thus, the multiplicative inverse of 17 modulo 23 is indeed \( \boxed{19} \).

Do you want any more details or have any questions?

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**Related Questions:**

1. What is the multiplicative inverse of 5 modulo 13?
2. How does the Extended Euclidean Algorithm work in general?
3. Can the multiplicative inverse of a number modulo \( n \) be found for non-prime \( n \)?
4. What are the properties of numbers with multiplicative inverses in a given modulus?
5. How does modular arithmetic apply in cryptography, such as in RSA?
6. What is the significance of the modular inverse in solving linear congruences?
7. Can the Chinese Remainder Theorem be used to find multiplicative inverses?
8. What happens if a number does not have a multiplicative inverse modulo \( n \)?

**Tip:** When verifying your results in modular arithmetic, it can be helpful to multiply the original number by the calculated inverse and ensure the result is congruent to 1 modulo the base.

Learn how to find the multiplicative inverse of 17 modulo 23 using the Extended Euclidean Algorithm. Detailed step-by-step solution and verification included. Suitable for advanced high school students and those interested in number theory.

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