Given a string of digits “0-9”. The task is find the number of substrings which are divisible by 8 but not by 3.
Input : str = "888" Output : 5 Substring indexes : (1, 1), (1, 2), (2, 2), (2, 3), (3, 3). Input : str = "6564525600" Output : 15
A number is divisible by 3 if sum of its digits is divisible by 3. And the number is divisible by 8 if last three digits are divisible by 8.
Now, the idea is to store the prefix sum of the string i.e count of prefixes such that sum of the digits of the prefix modulo 3 is either 0, 1, 2. Next we iterate over the string and for each position i, count the number of substrings ending at i and divisible by 8. From this value, we subtract the number of substrings ending at i and divisible by 3.
We define a |S| X 3 size 2D array, |S| is size of string, say dp[i][j].
dp[i][j] can be define as at any index i, number of substring starting from index 0 to index i having output j
when digits from index 0 are added upto index i and modulo 3. Therefore 0 <= j <= 2, since modulo 3. Now, we will iterate over string and check each one digit number, two digit number and three digit number which are divisible by 8. For one digit, just check if character at index is 8 or not. For two digit, check whether it is divisible by 8 and not divisible by 3. For three digit, form the number and check if it is divisible by 8 or not. If the number is divisible then last three digits must be divisible by 8. So all the substring ending at this index must be divisible by 8 i.e (i-3) substrings. But it will also contain those substring which are divisible by 3, so simply remove them using dp[i][j].
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Improved By : parashar