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# PHP Program to Count Trailing Zeroes in Factorial of a Number

## What is Factorial of a Number?

The factorial of a non-negative integer, denoted by the symbol "!", is the product of all positive integers less than or equal to that number. In other words, the factorial of a number is obtained by multiplying that number by all the positive integers below it.

For example, the factorial of 5 is calculated as:

5! = 5 x 4 x 3 x 2 x 1 = 120

Similarly, the factorial of 0 is defined to be 1:

0! = 1

Factorials are often used in mathematics and combinatorics to count permutations, combinations, and arrangements of objects. They also have applications in probability, calculus, and various other areas of mathematics.

## PHP Program to Count Trailing Zeroes in Factorial of a Number

**In the factorial of a number, trailing zeroes refer to the number of consecutive zeros at the end of the factorial's decimal representation.
**

For example 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Performing the multiplication

10! = 3,628,800

The factorial of 10 is 3,628,800.

Trailing zeroes in factorial of 10 are 2 because the number of consecutive zeros at the end of the factorial.

### Example

<?php function countTrailingZeroes($number) { $count = 0; // Divide the number by powers of 5 and count the quotient // The quotient represents the number of trailing zeroes while ($number >= 5) { $number = (int) ($number / 5); $count += $number; } return $count; } // Test the function $number = 20; $trailingZeroes = countTrailingZeroes($number); echo "The factorial of $number has $trailingZeroes trailing zeroes.<br>"; // Test the function $number = 14; $trailingZeroes = countTrailingZeroes($number); echo "The factorial of $number has $trailingZeroes trailing zeroes."; ?>

### Output

The factorial of 20 has 4 trailing zeroes. The factorial of 14 has 2 trailing zeroes.

### Explanation of code

In example code a PHP function is called countTrailingZeroes. This function calculates the number of trailing zeroes in the factorial of a given number. It does this by dividing the number by powers of 5 and counting the quotient. The while loop continues as long as the number is greater than or equal to 5. Within the loop, the number is divided by 5 using integer division to calculate the number of factors of 5 in the current number. The resulting quotient is added to a variable called $count, which keeps track of the count of trailing zeroes. After the loop finishes, the final count is returned from the function.

Below the function, there is a test case where the function is called with a value of 123. This calculates the number of trailing zeroes in the factorial of 20 using the countTrailingZeroes function. The result is stored in a variable called $trailingZeroes. Finally, the result is displayed using echo, providing the input number and the count of trailing zeroes in its factorial.

In this case, the factorial of 20 is 2,432,902,008,176,640,000 so the count of trailing zeroes in its factorial is 4 and the factorial of 14 is 87,178,291,200. So the count of trailing zeroes in its factorial is 2.

## Conclusion

The provided PHP program efficiently calculates the count of trailing zeroes in the factorial of a given number. It utilizes a while loop to divide the number by powers of 5 and count the quotient, representing the number of trailing zeroes. By leveraging this approach, the program avoids the need to compute the entire factorial. This technique is effective because trailing zeroes in a factorial result from factors of 5. Therefore, by counting the factors of 5, the program accurately determines the count of trailing zeroes. This code provides a convenient and efficient solution for calculating trailing zeroes in factorials, aiding in various mathematical and programming applications.