Number of substrings divisible by 6 in a string of integers
Last Updated :
23 Jul, 2025
Given a string consisting of integers 0 to 9. The task is to count the number of substrings which when convert into integer are divisible by 6. Substring does not contain leading zeroes. Examples:
Input : s = "606".
Output : 5
Substrings "6", "0", "6", "60", "606"
are divisible by 6.
Input : s = "4806".
Output : 5
"0", "6", "48", "480", "4806" are
substring which are divisible by 6.
Method 1: (Brute Force) The idea is to find all the substrings of the given string and check if substring is divisible by 6 or not.
Time Complexity: O(n2).
Method 2:(Dynamic Programming) As discussed in Check if a large number is divisible by 6 or not. A number is divisible by 6 if last digit is divisible by 2 and sum of digits is divisible by 3. The idea is to use Dynamic Programming, which enables us to compute answer quickly and efficiently by tracking previously computed answers and using these stored answer instead of recomputing values.
Let f(i, m) be the number of strings starting at index i and sum of their digits modulo 3 (so far) is m and number it represents is even. So, our answer would be {\displaystyle \sum_{i}^{n-1}f(i,0) }
Let x be the ith digit in the string. From f(i, m) we need to find all the even substrings that start in i + 1. Also, we will get an extra substring if (x + m) itself is divisible by 3 and x is even. So, we get recurrence relation
// We initially pass m (sum modulo 3 so far) as 0
f(i, m) = ((x + m)%3 == 0 and x%2 == 0) +
f(i + 1, (m + x)%3) // Recursive
By memorizing the states, we get O(n) solution. Below is implementation of this approach:
C++
// C++ program to calculate number of substring
// divisible by 6.
#include <bits/stdc++.h>
#define MAX 100002
using namespace std;
// Return the number of substring divisible by 6
// and starting at index i in s[] and previous sum
// of digits modulo 3 is m.
int f(int i, int m, char s[], int memoize[][3])
{
// End of the string.
if (i == strlen(s))
return 0;
// If already calculated, return the
// stored value.
if (memoize[i][m] != -1)
return memoize[i][m];
// Converting into integer.
int x = s[i] - '0';
// Increment result by 1, if current digit
// is divisible by 2 and sum of digits is
// divisible by 3.
// And recur for next index with new modulo.
int ans = ((x+m)%3 == 0 && x%2 == 0) +
f(i+1, (m+x)%3, s, memoize);
return memoize[i][m] = ans;
}
// Returns substrings divisible by 6.
int countDivBy6(char s[])
{
int n = strlen(s);
// For storing the value of all states.
int memoize[n+1][3];
memset(memoize, -1, sizeof memoize);
int ans = 0;
for (int i = 0; i < strlen(s); i++)
{
// If string contain 0, increment count by 1.
if (s[i] == '0')
ans++;
// Else calculate using recursive function.
// Pass previous sum modulo 3 as 0.
else
ans += f(i, 0, s, memoize);
}
return ans;
}
// Driven Program
int main()
{
char s[] = "4806";
cout << countDivBy6(s) << endl;
return 0;
}
// Java program to calculate number of substring
// divisible by 6.
import java.util.*;
class GFG
{
static int MAX = 100002;
// Return the number of substring divisible by 6
// and starting at index i in s[] and previous sum
// of digits modulo 3 is m.
static int f(int i, int m, char s[], int memoize[][])
{
// End of the string.
if (i == s.length)
{
return 0;
}
// If already calculated, return the
// stored value.
if (memoize[i][m] != -1)
{
return memoize[i][m];
}
// Converting into integer.
int x = s[i] - '0';
// Increment result by 1, if current digit
// is divisible by 2 and sum of digits is
// divisible by 3.
// And recur for next index with new modulo.
int ans = ((x + m) % 3 == 0 && x % 2 == 0) ? 1 + f(i + 1,
(m + x) % 3, s, memoize) : f(i + 1, (m + x) % 3, s, memoize);
memoize[i][m] = ans;
return memoize[i][m];
}
// Returns substrings divisible by 6.
static int countDivBy6(char s[])
{
int n = s.length;
// For storing the value of all states.
int[][] memoize = new int[n + 1][3];
for (int i = 0; i < n + 1; i++)
{
for (int j = 0; j < 3; j++)
{
memoize[i][j] = -1;
}
}
int ans = 0;
for (int i = 0; i < s.length; i++)
{
// If string contain 0, increment count by 1.
if (s[i] == '0')
{
ans++;
}
// Else calculate using recursive function.
// Pass previous sum modulo 3 as 0.
else
{
ans += f(i, 0, s, memoize);
}
}
return ans;
}
// Driven Program
public static void main(String[] args)
{
char s[] = "4806".toCharArray();
System.out.println(countDivBy6(s));
}
}
// This code is contributed by Rajput-Ji
# Python3 program to calculate number
# of substring
# Return the number of substring divisible
# by 6 and starting at index i in s[] and
# previous sum of digits modulo 3 is m.
def f(i, m, s, memoize):
# End of the string.
if (i == len(s)):
return 0
# If already calculated, return
# the stored value.
if (memoize[i][m] != -1):
return memoize[i][m]
# Converting into integer.
x = ord(s[i]) - ord('0')
# Increment result by 1, if current digit
# is divisible by 2 and sum of digits is
# divisible by 3.
# And recur for next index with new modulo.
ans = (((x + m) % 3 == 0 and x % 2 == 0) +
f(i + 1, (m + x) % 3, s, memoize))
memoize[i][m] = ans
return memoize[i][m]
# Returns substrings divisible by 6.
def countDivBy6(s):
n = len(s)
# For storing the value of all states.
memoize = [[-1] * 3 for i in range(n + 1)]
ans = 0
for i in range(len(s)):
# If string contain 0, increment
# count by 1.
if (s[i] == '0'):
ans += 1
# Else calculate using recursive function.
# Pass previous sum modulo 3 as 0.
else:
ans += f(i, 0, s, memoize)
return ans
# Driver Code
if __name__ == '__main__':
s = "4806"
print(countDivBy6(s))
# This code is contributed by PranchalK
// C# program to calculate number of substring
// divisible by 6.
using System;
class GFG
{
static int MAX = 100002;
// Return the number of substring divisible by 6
// and starting at index i in s[] and previous sum
// of digits modulo 3 is m.
static int f(int i, int m, char []s, int [,]memoize)
{
// End of the string.
if (i == s.Length)
{
return 0;
}
// If already calculated, return the
// stored value.
if (memoize[i,m] != -1)
{
return memoize[i,m];
}
// Converting into integer.
int x = s[i] - '0';
// Increment result by 1, if current digit
// is divisible by 2 and sum of digits is
// divisible by 3.
// And recur for next index with new modulo.
int ans = ((((x + m) % 3 == 0) && (x % 2 == 0)) ? 1 : 0)
+ f(i + 1, (m + x) % 3, s, memoize);
return memoize[i,m] = ans;
}
// Returns substrings divisible by 6.
static int countDivBy6(char []s)
{
int n = s.Length;
// For storing the value of all states.
int[,] memoize = new int[n + 1,3];
for (int i = 0; i < n + 1; i++)
{
for (int j = 0; j < 3; j++)
{
memoize[i,j] = -1;
}
}
int ans = 0;
for (int i = 0; i < s.Length; i++)
{
// If string contain 0, increment count by 1.
if (s[i] == '0')
{
ans++;
}
// Else calculate using recursive function.
// Pass previous sum modulo 3 as 0.
else
{
ans += f(i, 0, s, memoize);
}
}
return ans;
}
// Driver code
public static void Main(String[] args)
{
char []s = "4806".ToCharArray();
Console.WriteLine(countDivBy6(s));
}
}
/* This code is contributed by PrinciRaj1992 */
<script>
// JavaScript program to calculate number
// of substring
// Return the number of substring divisible
// by 6 and starting at index i in s[] and
// previous sum of digits modulo 3 is m.
function f(i, m, s, memoize){
// End of the string.
if (i == s.length)
return 0
// If already calculated, return
// the stored value.
if (memoize[i][m] != -1)
return memoize[i][m]
// Converting into integer.
let x = s.charCodeAt(i) - '0'.charCodeAt(0)
// Increment result by 1, if current digit
// is divisible by 2 and sum of digits is
// divisible by 3.
// And recur for next index with new modulo.
ans = (((x + m) % 3 == 0 && x % 2 == 0) + f(i + 1, (m + x) % 3, s, memoize))
memoize[i][m] = ans
return memoize[i][m]
}
// Returns substrings divisible by 6.
function countDivBy6(s){
let n = s.length
// For storing the value of all states.
let memoize = new Array(n + 1)
for(let i=0;i<n + 1;i++){
memoize[i] = new Array(3).fill(-1)
}
let ans = 0
for(let i=0;i<s.length;i++){
// If string contain 0, increment
// count by 1.
if (s[i] == '0')
ans += 1
// Else calculate using recursive function.
// Pass previous sum modulo 3 as 0.
else
ans += f(i, 0, s, memoize)
}
return ans
}
// Driver Code
let s = "4806"
document.write(countDivBy6(s))
// This code is contributed by shinjanpatra
</script>
Time Complexity: O(n).
Auxiliary Space: O(n)
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem