To find the number of trailing zeroes in a factorial, we focus on the factors of 10 present in the number. Since 10 is made of 2 and 5, and typically there are more factors of 2 than 5 in factorials, we only need to count the number of times 5 is a factor.
Steps to Calculate Trailing Zeroes
1. Identify the Factorial
Consider the factorial of a number \( n \), denoted as \( n! \).
2. Count the Factors of 5
Use the formula to count the number of factors of 5 in \( n! \):
- \( \text{Number of trailing zeroes} = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{5^2} \right\rfloor + \left\lfloor \frac{n}{5^3} \right\rfloor + \ldots \)
- Continue until \( 5^k \) where \( 5^k \leq n \).
Example Calculation
- To find the number of trailing zeroes in \( 100! \):
- Calculate \( \left\lfloor \frac{100}{5} \right\rfloor = 20 \)
- Calculate \( \left\lfloor \frac{100}{25} \right\rfloor = 4 \)
- \( \left\lfloor \frac{100}{125} \right\rfloor = 0 \) (stop here)
- Add them up: \( 20 + 4 = 24 \)
Thus, \( 100! \) has 24 trailing zeroes.
Key Points
- Focus on Factors of 5
Trailing zeroes arise from pairs of 2 and 5, but 2s are more frequent.
- Iterate Over Powers of 5
Always continue counting until the power of 5 exceeds \( n \).
Understanding these steps will help accurately determine the number of trailing zeroes in factorials, which is a common question in quantitative assessments.