To find the units digit of \( 2013! \) (2013 factorial), we must consider the prime factors involved in the factorial expression.

### Key Insight:

- \( 2013! \) represents the product of all integers from 1 to 2013. This includes many multiples of 10, since 10 is the product of 2 and 5.
- Specifically, every pair of a 2 and a 5 will produce a factor of 10, contributing a zero to the units digit.

### Analysis:

- There are clearly many multiples of 5 (like 5, 10, 15, 20, etc.) and multiples of 2 (like 2, 4, 6, 8, etc.) in \( 2013! \). For every multiple of 5, there is a corresponding multiple of 2, ensuring that the product has many factors of 10.
- Therefore, \( 2013! \) contains enough factors of 10 to make the units digit **0**.

### Conclusion:

The units digit of \( 2013! \) is **0**.

Would you like further details or have any other questions? Here are five related questions:

1. How do you calculate the number of trailing zeros in a factorial?
2. What is the highest power of 5 that divides \( 2013! \)?
3. Can the units digit of any factorial ever be a prime number?
4. How do prime factors affect the units digit of large factorials?
5. How would you compute the units digit of a smaller factorial like \( 9! \)?

**Tip**: For large numbers like factorials, focusing on patterns in prime factors can simplify finding properties like units digits.

Discover how to find the units digit of 2013 factorial (2013!) using prime factorization. This problem involves analyzing the number of factors of 10 in 2013! and explains why the units digit is zero. Ideal for high school students learning about factorials and number theory.

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