To find the multiplicative inverse of $32 \mod 97$, we need to find an integer $x$ such that:

$32x \equiv 1 \mod 97$

This can be done using the Extended Euclidean Algorithm, which helps solve equations of the form $ax + by = \gcd(a, b)$.

Since 32 and 97 are coprime ($\gcd(32, 97) = 1$), there exists an inverse. Here's the process:

1. Apply the Euclidean algorithm to find the greatest common divisor (gcd) of 32 and 97:

   $97 = 3 \times 32 + 1$ $32 = 32 \times 1 + 0$

   So, the gcd is $1$, confirming that an inverse exists.

2. Now, reverse the steps to express $1$ as a linear combination of $97$ and $32$:

   From $97 = 3 \times 32 + 1$, we can rearrange to: $1 = 97 - 3 \times 32$

3. Thus, $x = -3$ is the solution, but we need the inverse to be positive and in the range $0 \leq x < 97$. To do this, add 97 to $-3$:

   $x = -3 + 97 = 94$

Therefore, the multiplicative inverse of $32 \mod 97$ is $\boxed{94}$.

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Would you like more details on how the Extended Euclidean Algorithm works, or do you have any other questions?

Here are 5 related questions for further exploration:

1. What is the Euclidean Algorithm, and how is it used in finding gcd?
2. How does the Extended Euclidean Algorithm differ from the standard Euclidean Algorithm?
3. What is the importance of the multiplicative inverse in modular arithmetic?
4. Can all numbers have multiplicative inverses modulo $n$?
5. How would the process differ if the modulus was a composite number?

Tip: When finding an inverse, always ensure the number and modulus are coprime (gcd = 1), otherwise, an inverse does not exist.