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 8function 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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