Step by Step solution about How to resolve the algorithm Sum multiples of 3 and 5 step by step in the PHP programming language

The provided PHP code is designed to calculate the sum of all multiples of 3 or 5 below a specified maximum value. It utilizes three different approaches for this calculation. Here's a detailed explanation of each approach:

1. Using a Simple Loop:

In the first code block, a straightforward loop is employed to iterate through numbers from 1 to the specified maximum value (excluding the maximum itself). For each number, it checks if it's divisible by 3 or 5 (i.e., if the remainder is 0 when divided by 3 or 5). If the condition is met, the number is added to a running sum. Finally, the total sum is printed to the console.

2. Using Helper Functions:

In the second code block, two helper functions are defined:

• sum_multiples($max, $divisor): This function calculates the sum of all multiples of a given divisor ($divisor) up to a specified maximum value ($max). It uses a mathematical formula to compute the sum based on the number of multiples and the divisor.
• sum_multiples($max, $divisor): This function calculates the sum of all multiples of a given divisor ($divisor) up to a specified maximum value ($max). It uses a mathematical formula to compute the sum based on the number of multiples and the divisor.

The main code then calculates the sum of multiples for divisors 3 and 5 up to the specified maximum and subtracts the sum of multiples for divisor 15 (which overlaps with both 3 and 5). The resulting sum is printed to the console.

3. Using GMP Extension for Large Numbers:

In the third code block, the popular GMP (GNU Multiple Precision) extension is utilized to handle large numbers efficiently. The sum_multiples_gmp($max, $divisor) function is similar to the sum_multiples function, but it uses GMP functions for arithmetic operations on large integers.

The main code iterates through a range of maximum values and calculates the sum of multiples for divisors 3 and 5 up to each maximum. It then subtracts the sum of multiples for divisor 15. The results are printed to the console, showcasing the performance of the GMP extension in handling large numbers.

$max = 1000;
$sum = 0;
for ($i = 1 ; $i < $max ; $i++) {
    if (($i % 3 == 0) or ($i % 5 == 0)) {
        $sum += $i;
    }
}
echo $sum, PHP_EOL;


function sum_multiples($max, $divisor) {
    // Number of multiples of $divisor <= $max
    $num = floor($max / $divisor);
    // Sum of multiples of $divisor
    return ($divisor * $num * ($num + 1) / 2);
}

$max = 1000;
$sum = sum_multiples($max - 1,  3)
     + sum_multiples($max - 1,  5)
     - sum_multiples($max - 1, 15);
echo $sum, PHP_EOL;


function sum_multiples_gmp($max, $divisor) {
    // Number of multiples of $divisor <= $max
    $num = gmp_div($max, $divisor);
    // Sum of multiples of $divisor
    return gmp_div(gmp_mul(gmp_mul($divisor, $num), gmp_add($num, 1)), 2);
}

for ($i = 0, $n = gmp_init(10) ; $i < 21 ; $i++, $n = gmp_mul($n, 10)) {
    $max = gmp_sub($n, 1);
    $sum = 
        gmp_sub(
            gmp_add(
                sum_multiples_gmp($max, 3),
                sum_multiples_gmp($max, 5)
            ),
            sum_multiples_gmp($max, 15)
        );
    printf('%22s : %s' . PHP_EOL, gmp_strval($n), $sum);
}