Php Program to Count rotations divisible by 8

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Given a large positive number as string, count all rotations of the given number which are divisible by 8.

Examples:

Input: 8
Output: 1

Input: 40
Output: 1
Rotation: 40 is divisible by 8
          04 is not divisible by 8

Input : 13502
Output : 0
No rotation is divisible by 8

Input : 43262488612
Output : 4

Approach: For large numbers it is difficult to rotate and divide each number by 8. Therefore, ‘divisibility by 8’ property is used which says that a number is divisible by 8 if the last 3 digits of the number is divisible by 8. Here we do not actually rotate the number and check last 8 digits for divisibility, instead we count consecutive sequence of 3 digits (in circular way) which are divisible by 8.

Illustration:

Consider a number 928160
Its rotations are 928160, 092816, 609281, 
160928, 816092, 281609.
Now form consecutive sequence of 3-digits from 
the original number 928160 as mentioned in the 
approach. 
3-digit: (9, 2, 8), (2, 8, 1), (8, 1, 6), 
(1, 6, 0),(6, 0, 9), (0, 9, 2)
We can observe that the 3-digit number formed by 
the these sets, i.e., 928, 281, 816, 160, 609, 092, 
are present in the last 3 digits of some rotation.
Thus, checking divisibility of these 3-digit numbers
gives the required number of rotations.

PHP

<?php
// PHP program to count all 
// rotations divisible by 8
 
// function to count of all 
// rotations divisible by 8
function countRotationsDivBy8($n)
{
    $len = strlen($n);
    $count = 0;
 
    // For single digit number
    if ($len == 1) 
    {
        $oneDigit = $n[0] - '0';
        if ($oneDigit % 8 == 0)
            return 1;
        return 0;
    }
 
    // For two-digit numbers 
    // (considering all pairs)
    if ($len == 2) 
    {
 
        // first pair
        $first = ($n[0] - '0') * 10 + 
                 ($n[1] - '0');
 
        // second pair
        $second = ($n[1] - '0') * 10 + 
                  ($n[0] - '0');
 
        if ($first % 8 == 0)
            $count++;
        if ($second % 8 == 0)
            $count++;
        return $count;
    }
 
    // considering all 
    // three-digit sequences
    $threeDigit;
    for ($i = 0; $i < ($len - 2); $i++) 
    {
        $threeDigit = ($n[$i] - '0') * 100 + 
                      ($n[$i + 1] - '0') * 10 + 
                      ($n[$i + 2] - '0');
        if ($threeDigit % 8 == 0)
            $count++;
    }
 
    // Considering the number 
    // formed by the last digit
    // and the first two digits
    $threeDigit = ($n[$len - 1] - '0') * 100 + 
                  ($n[0] - '0') * 10 + 
                  ($n[1] - '0');
 
    if ($threeDigit % 8 == 0)
        $count++;
 
    // Considering the number 
    // formed by the last two
    // digits and the first digit
    $threeDigit = ($n[$len - 2] - '0') * 100 + 
                  ($n[$len - 1] - '0') * 10 + 
                   ($n[0] - '0');
    if ($threeDigit % 8 == 0)
        $count++;
 
    // required count 
    // of rotations
    return $count;
}
 
// Driver Code
$n = "43262488612";
echo "Rotations: " . 
      countRotationsDivBy8($n);
 
// This code is contributed by mits.
?>

Output:

Rotations: 4

Time Complexity : O(n), where n is the number of digits in input number.
Auxiliary Space: O(1)

Please refer complete article on Count rotations divisible by 8 for more details!

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