Current Location: Home> Latest Articles> How to Calculate the Number of Trailing Zeros in a Factorial Using PHP: A PHP Code Example

How to Calculate the Number of Trailing Zeros in a Factorial Using PHP: A PHP Code Example

gitbox 2025-06-17

What is a Factorial?

A factorial is an important concept in mathematics that represents the product of all positive integers less than or equal to a given number. For example, 5 factorial (5!) is represented as: 5 × 4 × 3 × 2 × 1 = 120. Factorials are widely used in fields such as combinatorics, statistics, and mathematical analysis.

What is the Number of Trailing Zeros?

The number of trailing zeros refers to how many zeros are at the end of a number. For a factorial, the number of trailing zeros is determined primarily by the number of 5s in its prime factorization. This is because, during the multiplication of numbers in a factorial, the factor of 2 is more frequent than the factor of 5. Thus, the number of trailing zeros is equivalent to the number of 5s in the prime factorization. For example, the factorial of 5265 has 1315 trailing zeros.

PHP Programming Implementation

Below is a PHP program to calculate the number of trailing zeros in the factorial of a number:


// Define a function to calculate trailing zeros
function countZeroesInFactorial($x) {
    $zeroes = 0;
    for ($i = 5; $i <= $x; $i *= 5) {
        $zeroes += floor($x / $i);
    }
    return $zeroes;
}

// Test
$num = 10;
$result = countZeroesInFactorial($num);
echo "The number of trailing zeroes in factorial of $num is: $result";

Code Explanation

The `countZeroesInFactorial` function in the code above takes a parameter `$x` and calculates the number of trailing zeros in `$x!`. Inside the function, a variable `$zeroes` is used to store the count of trailing zeros. The loop starts from 5, as the number of factors of 2 in a factorial is always greater than that of 5. In each iteration, the loop multiplies `$i` by 5 and calculates how many factors of 5 exist in `$x`, accumulating the result in `$zeroes`.

In the test section, a number `$num` is assigned a value, and the function `countZeroesInFactorial` is called to calculate the result. Finally, the result is printed using `echo`.

Running Result

The output of the code is as follows:


The number of trailing zeroes in factorial of 10 is: 2

After running the code, the output shows the number of trailing zeros in the factorial of 10. In this example, the factorial of 10 is 3628800, and it has 2 trailing zeros, which matches the expected result.

Conclusion

This article explained the concept of factorials and how to calculate the number of trailing zeros in a factorial. We provided a PHP code example, a detailed code explanation, and the running results to help readers better understand how trailing zeros are determined in factorials. We hope this article helps you understand factorial calculations and how to compute the number of trailing zeros in PHP programming.

Related