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

What is factorial?

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 as 1:

0! = 1

Facttorials 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 zeros refer to the number of consecutive zeros in the decimal representation of the factorial.

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

Perform multiplication operation

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.

Code explanation

A PHP function named countTrailingZeroes is called in the sample code. This function counts the number of trailing zeros in the factorial of a given number. It does this by dividing the number by a power of 5 and calculating the quotient. As long as the number is greater than or equal to 5, the while loop will continue executing. Inside the loop, the number is divided by 5 using integer division to count the number of factors of 5 in the current number. Add the resulting quotient to a variable called $count, which keeps track of the number of trailing zeros. After the loop ends, return the final count value from the function.

Below the function, there is a test case where the function is called with a value of 123. This test case counts the number of trailing zeros in the factorial of 20 using the countTrailingZeroes function. The results are stored in a variable called $trailingZeroes. Finally, use echo to display the result, providing the input number and the number of trailing zeros in its factorial

In this case, the factorial of 20 is 2,432,902,008,176,640,000, so its factorial has 4 zeros at the end, while the factorial of 14 is 87,178,291,200. So it has 2 zeros at the end of its factorial.

Conclusion

Provided PHP program to efficiently calculate the number of trailing zeros in the factorial of a given number. It utilizes a while loop to divide the number by a power of 5 and calculates the quotient, representing the number of trailing zeros. By utilizing this approach, the program avoids the need to calculate the entire factorial. This technique works because the trailing zeros in the factorial come from the factor 5. Therefore, by calculating factors of 5, the program can accurately determine the number of trailing zeros. This code provides a convenient and efficient solution for computing trailing zeros in factorials, useful for a variety of mathematical and programming applications.

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