Trailing Zeros in n Factorial

Count decimal trailing zeros of n! exactly by counting factors of five, accepting nonnegative decimal integers up to 200 characters.

Description

Decimal trailing zeros in a factorial come from paired factors of two and five.

Decimal trailing zeros in a factorial come from paired factors of two and five. Factors of five are the limiting supply, including extra factors from powers of five.^[1]

The tool returns zeros. The indexing and output conventions below determine how to read these values.

When to use Trailing Zeros in n Factorial

Use this for the following task: Count decimal trailing zeros of n! exactly by counting factors of five, accepting nonnegative decimal integers up to 200 characters. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Trailing Zeros in n Factorial and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n according to the field table, then run the tool.
Read zeros. Preserve exact decimal strings and the stated indexing or coefficient order.
Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.

Input field	Meaning	Value type and bounds	Requirement
n	Nonnegative factorial argument	string (integer/arbitrary); maximum text length 200; pattern ^(0\|[1-9][0-9]*)$	Required

Parse n as BigInt and repeatedly divide by five, adding each positive quotient until it becomes zero. Return an exact decimal count.

zeros(n!)=Σ[j≥1]floor(n/5^j)

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
zeros	Number of trailing decimal zeros	string (integer/arbitrary)

json

{
  "input": {
    "n": "100"
  },
  "output": {
    "zeros": "24"
  }
}

Limitations and assumptions

The factorial itself is never constructed. The tool checks base ten only; n=0 returns zero trailing zeros because 0!=1.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Factorion Checker when you need its operation: Check whether a nonnegative decimal integer equals the sum of factorials of its digits, including 0!=1 and an input resource limit of 200 characters. Choose Euler’s Number from the Factorial Series for the complementary operation: Approximate e by summing the first requested terms of Σ1/k!, beginning with k=0, using compensated floating-point addition and a 10,000-term resource limit.

Background reference: OEIS A027868.

References

OEIS A027868 — OEIS Foundation

Similar or alternative tools

Factorion Checker
Check whether a nonnegative decimal integer equals the sum of factorials of its digits, including 0!=1 and an input resource limit of 200 characters.
Euler’s Number from the Factorial Series
Approximate e by summing the first requested terms of Σ1/k!, beginning with k=0, using compensated floating-point addition and a 10,000-term resource limit.