To compute $4! \mod 5$, follow these steps:

1. Calculate $4!$: $4! = 4 \times 3 \times 2 \times 1 = 24$

2. Reduce $24 \mod 5$: Divide $24$ by $5$ and find the remainder: $24 \div 5 = 4 \text{ remainder } 4$

   Thus, $24 \equiv 4 \pmod{5}$

Final Answer:

$4! \mod 5 = 4$

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Related Questions:

1. How do you calculate factorials for larger numbers modulo another number?
2. Why is modular arithmetic important in number theory?
3. What is the relationship between factorials and prime moduli?
4. How can modular arithmetic be used in cryptography?
5. What are the properties of factorials under modular arithmetic?

Tip:

For modular arithmetic, you can simplify intermediate results during calculations to avoid working with large numbers.