85! - Factorial of 85

85! = 85 × 84 × 83 × 82 × ⋯

85 factorial has 129 digits. The number of zeros at the end is 20.

2817104114 3805502769 4947944226 0611594800 5663433057 4206405101 9127525600 2615979593 3451040286 4523409240 1827512320 0000000000 000000000

Factorial of eighty-five

Instructions

Enter an integer 0-50,000. The calculator will compute the factorial and the number of digits it contains.

What is a factorial?

A factorial of N is the product of all positive integers between 1 and N inclusive. For example, the factorial of 5 is 5×4×3×2×1=120. The one exception is 0!, which is defined as 1. Factorials of negative numbers are undefined.

$$( n! = \begin{cases} 1 & \text{if $n=0$} \\ 1 & \text{if $n=1$} \\ n \times (n - 1)! & \text{if $n>1$} \end{cases})$$

Factorials are common in various branches of mathematics including combinatorics, number theory and taylor expansions.

Number of Trailing Zeros

To determine the number of zeros at the end of a factorial, recursively divide the number by 5 until the quotient is less than 5, and sum the results after applying the greatest integer function.

The greatest integer function (usually denoted by brackets) is the rounded down integer of a value. For example, [5] = 5, [4.5] = 4, [-4.5] = -5.

For example, the number of trailing zeros in 85! is ([85/5]=17) + ([17/5]=3) = 20.

Digits in 85!

Digit Count
036 (27.91%)
113 (10.08%)
214 (10.85%)
38 (6.2%)
415 (11.63%)
513 (10.08%)
69 (6.98%)
77 (5.43%)
85 (3.88%)
99 (6.98%)
Total129