To find the multiplicative inverse of $97!$ modulo $103$, we need to solve for $x$ in the equation:

$97! \cdot x \equiv 1 \pmod{103}$

This means $x$ is the multiplicative inverse of $97!$ modulo $103$. We can find it using Fermat's Little Theorem, which is applicable because $103$ is a prime number.

Step-by-step process:

1. Fermat's Little Theorem: For any integer $a$ and a prime $p$, Fermat's Little Theorem states that: $a^{p-1} \equiv 1 \pmod{p}$ which implies: $a^{p-2} \equiv a^{-1} \pmod{p}$ Thus, the inverse of $97!$ modulo $103$ is: $(97!)^{101} \pmod{103}$

2. Simplifying using Wilson's Theorem: Wilson's theorem states that for a prime $p$, $(p-1)! \equiv -1 \pmod{p}$ Therefore, for $p = 103$, $102! \equiv -1 \pmod{103}$ Since $102! = 102 \cdot 101 \cdot 100 \cdot 99 \cdot 98 \cdot 97!$, we can break it down: $102! = 102 \cdot 101 \cdot 100 \cdot 99 \cdot 98 \cdot 97!$ We can simplify modulo $103$:

   • $102 \equiv -1 \pmod{103}$
   • $101 \equiv -2 \pmod{103}$
   • $100 \equiv -3 \pmod{103}$
   • $99 \equiv -4 \pmod{103}$
   • $98 \equiv -5 \pmod{103}$

   So, $102! \equiv (-1)(-2)(-3)(-4)(-5) \cdot 97! \equiv (-1)^5 \cdot 120 \cdot 97! \equiv -120 \cdot 97! \pmod{103}$ We know that $102! \equiv -1 \pmod{103}$, thus: $-120 \cdot 97! \equiv -1 \pmod{103}$ Multiplying both sides by $-1$: $120 \cdot 97! \equiv 1 \pmod{103}$

3. Finding the inverse of 120 modulo 103: To isolate $97!$, we need the inverse of $120 \pmod{103}$. Using the extended Euclidean algorithm:

   • $120 = 1 \cdot 103 + 17$
   • $103 = 6 \cdot 17 + 1$
   • $17 = 17 \cdot 1 + 0$

   Reversing the steps, we get: $1 = 103 - 6 \cdot 17 = 103 - 6 \cdot (120 - 1 \cdot 103) = 7 \cdot 103 - 6 \cdot 120$ Thus, $-6 \equiv 97 \pmod{103}$, meaning the inverse of $120$ modulo $103$ is $97$.

4. Conclusion: The multiplicative inverse of $97!$ modulo $103$ is: $97$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you use Fermat's Little Theorem to find inverses modulo primes?
2. What are the applications of Wilson's Theorem in number theory?
3. How does the extended Euclidean algorithm help in finding modular inverses?
4. Why is it important to compute factorials modulo primes efficiently?
5. How would this method change for a non-prime modulus?

Tip: Always check if a modulus is prime before applying Fermat's Little Theorem or Wilson's Theorem.