Factorial Trailing Zeros Calculator

This factorial trailing zeros calculator counts how many zeros sit at the end of a factorial without ever computing the (astronomically large) factorial itself. Enter n and the calculator returns the number of trailing zeros of n factorial, along with the number of factors of 5 and of 2 it contains. A trailing zero comes from a factor of 10, which is a 5 times a 2. Since factors of 2 are far more common than factors of 5 in a factorial, the number of trailing zeros equals the number of factors of 5, found by adding up n divided by 5, by 25, by 125 and so on. The formula, a worked example and the assumptions are set out below.

n (0 to 100,000,000)

Factors of 5

Factors of 2

The formula

The number of trailing zeros of n! is the sum of the floors of n / 5, n / 25, n / 125 and so on, until the term is zero. This counts the factors of 5, which is the limiting factor for making tens.

Worked example

For 100!: floor(100/5) = 20, floor(100/25) = 4, floor(100/125) = 0, so there are 20 + 4 = 24 trailing zeros. Enter 100 to confirm.

Frequently asked questions

Why count factors of 5, not 10?

A trailing zero needs a factor of 10, which is 5 times 2. Factors of 2 are plentiful, so the number of 5s sets the limit.

How many zeros does 100! end in?

24, from 20 multiples of 5 plus 4 extra from multiples of 25.

Do I need the actual factorial?

No. The count uses only divisions of n, so it works for huge n instantly.

Who this calculator is for

This calculator is for students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard formula is applied.
Results are rounded for display.

Formula and sources

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