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.
// C++ program to count all rotations divisible // by 8 #include <bits/stdc++.h> using namespace std;
// function to count of all rotations divisible // by 8 int countRotationsDivBy8(string n)
{ int len = n.length();
int count = 0;
// For single digit number
if (len == 1) {
int oneDigit = n[0] - '0' ;
if (oneDigit % 8 == 0)
return 1;
return 0;
}
// For two-digit numbers (considering all
// pairs)
if (len == 2) {
// first pair
int first = (n[0] - '0' ) * 10 + (n[1] - '0' );
// second pair
int 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
int threeDigit;
for ( int 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 program to test above int main()
{ string n = "43262488612" ;
cout << "Rotations: "
<< countRotationsDivBy8(n);
return 0;
} |
Output:
Rotations: 4
Time Complexity: O(n), where n is the number of digits in the input number.
Auxiliary Space: O(1)
Please refer complete article on Count rotations divisible by 8 for more details!