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Count numbers divisible by K in a range with Fibonacci digit sum for Q queries
• Difficulty Level : Medium
• Last Updated : 21 Apr, 2020

Given an array arr[][] containing Q queries and an integer K where each query consists of a range [L, R], the task is to find the count of integers in the given range whose digit sum is a Fibonacci number and divisible by K.

Examples:

Input: arr[][] = { {1, 11}, {5, 15}, {2, 24} }, K = 2
Output: 3 2 5
Explanation:
For query 1: 2, 8 and 11 are the numbers in the given range whose digit sum is Fibonacci and divisible by K.
For query 2: 8 and 11 are the numbers in the given range whose digit sum is Fibonacci and divisible by K.
For query 3: 2, 8, 11, 17 and 20 are the numbers in the given range whose digit sum is Fibonacci and divisible by K.

Input: arr[][] = { {2, 17}, {3, 24} }, K = 3
Output: 4 4
Explanation:
For query 1: 2, 8, 11 and 17 are the numbers in the given range whose digit sum is Fibonacci and divisible by K.
For query 2: 8, 11, 17 and 20 are the numbers in the given range whose digit sum is Fibonacci and divisible by K.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The idea is to use hashing to precompute and store the Fibonacci nodes up to the maximum value in the given range, to make the checking easy and efficient (in O(1) time).

• After precomputation, mark all the integers from 1 to maxVal which are divisible by K and are fibonacci.
• Find the prefix sum of the marked array.
• Answer the given queries by prefix[right] – prefix[left – 1].

Below is the implementation of the above approach:

## C/C++

 `// C++ program to count the integers ` `// in a range [L, R] such that ` `// their digit sum is Fibonacci ` `// and divisible by K ` ` `  `#include ` `using` `namespace` `std; ` ` `  `const` `int` `maxSize = 1e5 + 5; ` `bool` `isFib[maxSize]; ` `int` `prefix[maxSize]; ` ` `  `// Function to return the ` `// digit sum of a number ` `int` `digitSum(``int` `num) ` `{ ` `    ``int` `s = 0; ` `    ``while` `(num != 0) { ` `        ``s = s + num % 10; ` `        ``num = num / 10; ` `    ``} ` `    ``return` `s; ` `} ` ` `  `// Function to generate all the Fibonacci ` `// numbers upto maxSize ` `void` `generateFibonacci() ` `{ ` `    ``memset``(isFib, ``false``, ``sizeof``(isFib)); ` ` `  `    ``// Adding the first two Fibonacci ` `    ``// numbers in the set ` `    ``int` `prev = 0, curr = 1; ` `    ``isFib[prev] = isFib[curr] = ``true``; ` ` `  `    ``// Computing the remaining Fibonacci ` `    ``// numbers based on the previous ` `    ``// two Fibonacci numbers ` `    ``while` `(curr < maxSize) { ` `        ``int` `temp = curr + prev; ` `        ``isFib[temp] = ``true``; ` `        ``prev = curr; ` `        ``curr = temp; ` `    ``} ` `} ` ` `  `// Pre-Computation till maxSize ` `// and for a given K ` `void` `precompute(``int` `k) ` `{ ` `    ``generateFibonacci(); ` ` `  `    ``for` `(``int` `i = 1; i < maxSize; i++) { ` ` `  `        ``// Getting the digit sum ` `        ``int` `sum = digitSum(i); ` ` `  `        ``// Check if the digit sum ` `        ``// is Fibonacci and divisible by k ` `        ``if` `(isFib[sum] == ``true` `            ``&& sum % k == 0) { ` `            ``prefix[i]++; ` `        ``} ` `    ``} ` ` `  `    ``// Taking Prefix Sum ` `    ``for` `(``int` `i = 1; i < maxSize; i++) { ` `        ``prefix[i] = prefix[i] ` `                    ``+ prefix[i - 1]; ` `    ``} ` `} ` ` `  `// Function to perform the queries ` `void` `performQueries( ` `    ``int` `k, ``int` `q, ` `    ``vector >& query) ` `{ ` `    ``// Precompute the results ` `    ``precompute(k); ` ` `  `    ``vector<``int``> ans; ` ` `  `    ``// Iterating through the queries ` `    ``for` `(``int` `i = 0; i < q; i++) { ` ` `  `        ``int` `l = query[i], ` `            ``r = query[i]; ` ` `  `        ``// Getting count of range ` `        ``// in range [L, R] ` `        ``int` `cnt = prefix[r] ` `                  ``- prefix[l - 1]; ` `        ``cout << cnt << endl; ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``vector > query ` `        ``= { { 1, 11 }, ` `            ``{ 5, 15 }, ` `            ``{ 2, 24 } }; ` `    ``int` `k = 2, q = query.size(); ` ` `  `    ``performQueries(k, q, query); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to count the integers ` `// in a range [L, R] such that ` `// their digit sum is Fibonacci ` `// and divisible by K ` `import` `java.util.*; ` ` `  `class` `GFG{ ` `  `  `static` `int` `maxSize = (``int``) (1e5 + ``5``); ` `static` `boolean` `[]isFib  = ``new` `boolean``[maxSize]; ` `static` `int` `[]prefix = ``new` `int``[maxSize]; ` `  `  `// Function to return the ` `// digit sum of a number ` `static` `int` `digitSum(``int` `num) ` `{ ` `    ``int` `s = ``0``; ` `    ``while` `(num != ``0``) { ` `        ``s = s + num % ``10``; ` `        ``num = num / ``10``; ` `    ``} ` `    ``return` `s; ` `} ` `  `  `// Function to generate all the Fibonacci ` `// numbers upto maxSize ` `static` `void` `generateFibonacci() ` `{ ` `    ``Arrays.fill(isFib, ``false``); ` `  `  `    ``// Adding the first two Fibonacci ` `    ``// numbers in the set ` `    ``int` `prev = ``0``, curr = ``1``; ` `    ``isFib[prev] = isFib[curr] = ``true``; ` `  `  `    ``// Computing the remaining Fibonacci ` `    ``// numbers based on the previous ` `    ``// two Fibonacci numbers ` `    ``while` `(curr < maxSize) { ` `        ``int` `temp = curr + prev; ` `        ``if``(temp < maxSize) ` `            ``isFib[temp] = ``true``; ` `        ``prev = curr; ` `        ``curr = temp; ` `    ``} ` `} ` `  `  `// Pre-Computation till maxSize ` `// and for a given K ` `static` `void` `precompute(``int` `k) ` `{ ` `    ``generateFibonacci(); ` `  `  `    ``for` `(``int` `i = ``1``; i < maxSize; i++) { ` `  `  `        ``// Getting the digit sum ` `        ``int` `sum = digitSum(i); ` `  `  `        ``// Check if the digit sum ` `        ``// is Fibonacci and divisible by k ` `        ``if` `(isFib[sum] == ``true` `            ``&& sum % k == ``0``) { ` `            ``prefix[i]++; ` `        ``} ` `    ``} ` `  `  `    ``// Taking Prefix Sum ` `    ``for` `(``int` `i = ``1``; i < maxSize; i++) { ` `        ``prefix[i] = prefix[i] ` `                    ``+ prefix[i - ``1``]; ` `    ``} ` `} ` `  `  `// Function to perform the queries ` `static` `void` `performQueries( ` `    ``int` `k, ``int` `q, ` `    ``int``[][] query) ` `{ ` `    ``// Precompute the results ` `    ``precompute(k); ` ` `  `    ``// Iterating through the queries ` `    ``for` `(``int` `i = ``0``; i < q; i++) { ` `  `  `        ``int` `l = query[i][``0``], ` `            ``r = query[i][``1``]; ` `  `  `        ``// Getting count of range ` `        ``// in range [L, R] ` `        ``int` `cnt = prefix[r] ` `                  ``- prefix[l - ``1``]; ` `        ``System.out.print(cnt +``"\n"``); ` `    ``} ` `} ` `  `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` ` `  `    ``int` `[][]query ` `        ``= { { ``1``, ``11` `}, ` `            ``{ ``5``, ``15` `}, ` `            ``{ ``2``, ``24` `} }; ` `    ``int` `k = ``2``, q = query.length; ` `  `  `    ``performQueries(k, q, query); ` `} ` `} ` ` `  `// This code is contributed by Princi Singh `

## Python3

 `# Python 3 program to count the integers ` `# in a range [L, R] such that ` `# their digit sum is Fibonacci ` `# and divisible by K ` `maxSize ``=` `100005` `isFib ``=` `[``False``]``*``(maxSize) ` `prefix ``=` `[``0``]``*``maxSize ` ` `  `# Function to return the ` `# digit sum of a number ` `def` `digitSum(num): ` `    ``s ``=` `0` `    ``while` `(num !``=` `0``): ` `        ``s ``=` `s ``+` `num ``%` `10` `        ``num ``=` `num ``/``/` `10` `     `  `    ``return` `s ` ` `  `# Function to generate all the Fibonacci ` `# numbers upto maxSize ` `def` `generateFibonacci(): ` ` `  `    ``global` `isFib ` ` `  `    ``# Adding the first two Fibonacci ` `    ``# numbers in the set ` `    ``prev ``=` `0` `    ``curr ``=` `1` `    ``isFib[prev] ``=` `True` `    ``isFib[curr] ``=` `True` ` `  `    ``# Computing the remaining Fibonacci ` `    ``# numbers based on the previous ` `    ``# two Fibonacci numbers ` `    ``while` `(curr < maxSize): ` `        ``temp ``=` `curr ``+` `prev ` `        ``if` `temp < maxSize: ` `            ``isFib[temp] ``=` `True` `        ``prev ``=` `curr ` `        ``curr ``=` `temp ` ` `  `# Pre-Computation till maxSize ` `# and for a given K ` `def` `precompute(k): ` ` `  `    ``generateFibonacci() ` `    ``global` `prefix ` `     `  `    ``for` `i ``in` `range``(``1``, maxSize): ` ` `  `        ``# Getting the digit sum ` `        ``sum` `=` `digitSum(i) ` ` `  `        ``# Check if the digit sum ` `        ``# is Fibonacci and divisible by k ` `        ``if` `(isFib[``sum``] ``=``=` `True` `            ``and` `sum` `%` `k ``=``=` `0``): ` `            ``prefix[i] ``+``=` `1` ` `  `    ``# Taking Prefix Sum ` `    ``for` `i ``in` `range``(``1``, maxSize): ` `        ``prefix[i] ``=` `prefix[i]``+` `prefix[i ``-` `1``] ` ` `  `# Function to perform the queries ` `def` `performQueries(k, q,query): ` `     `  `    ``# Precompute the results ` `    ``precompute(k) ` ` `  `    ``# Iterating through the queries ` `    ``for` `i ``in` `range``(q): ` ` `  `        ``l ``=` `query[i][``0``] ` `        ``r ``=` `query[i][``1``] ` ` `  `        ``# Getting count of range ` `        ``# in range [L, R] ` `        ``cnt ``=` `prefix[r]``-` `prefix[l ``-` `1``] ` `        ``print``(cnt) ` ` `  `# Driver code ` `if` `__name__ ``=``=` `"__main__"``: ` `    ``query ``=` `[ [ ``1``, ``11` `], ` `            ``[ ``5``, ``15` `], ` `            ``[ ``2``, ``24` `] ] ` `    ``k ``=` `2` `    ``q ``=` `len``(query) ` ` `  `    ``performQueries(k, q, query) ` ` `  `# This code is contributed by chitranayal `

## C#

 `// C# program to count the integers ` `// in a range [L, R] such that ` `// their digit sum is Fibonacci ` `// and divisible by K ` `using` `System; ` ` `  `class` `GFG{ ` `   `  `static` `int` `maxSize = (``int``) (1e5 + 5); ` `static` `bool` `[]isFib  = ``new` `bool``[maxSize]; ` `static` `int` `[]prefix = ``new` `int``[maxSize]; ` `   `  `// Function to return the ` `// digit sum of a number ` `static` `int` `digitSum(``int` `num) ` `{ ` `    ``int` `s = 0; ` `    ``while` `(num != 0) { ` `        ``s = s + num % 10; ` `        ``num = num / 10; ` `    ``} ` `    ``return` `s; ` `} ` `   `  `// Function to generate all the Fibonacci ` `// numbers upto maxSize ` `static` `void` `generateFibonacci() ` `{ ` ` `  `    ``// Adding the first two Fibonacci ` `    ``// numbers in the set ` `    ``int` `prev = 0, curr = 1; ` `    ``isFib[prev] = isFib[curr] = ``true``; ` `   `  `    ``// Computing the remaining Fibonacci ` `    ``// numbers based on the previous ` `    ``// two Fibonacci numbers ` `    ``while` `(curr < maxSize) { ` `        ``int` `temp = curr + prev; ` `        ``if``(temp < maxSize) ` `            ``isFib[temp] = ``true``; ` `        ``prev = curr; ` `        ``curr = temp; ` `    ``} ` `} ` `   `  `// Pre-Computation till maxSize ` `// and for a given K ` `static` `void` `precompute(``int` `k) ` `{ ` `    ``generateFibonacci(); ` `   `  `    ``for` `(``int` `i = 1; i < maxSize; i++) { ` `   `  `        ``// Getting the digit sum ` `        ``int` `sum = digitSum(i); ` `   `  `        ``// Check if the digit sum ` `        ``// is Fibonacci and divisible by k ` `        ``if` `(isFib[sum] == ``true` `            ``&& sum % k == 0) { ` `            ``prefix[i]++; ` `        ``} ` `    ``} ` `   `  `    ``// Taking Prefix Sum ` `    ``for` `(``int` `i = 1; i < maxSize; i++) { ` `        ``prefix[i] = prefix[i] ` `                    ``+ prefix[i - 1]; ` `    ``} ` `} ` `   `  `// Function to perform the queries ` `static` `void` `performQueries( ` `    ``int` `k, ``int` `q, ` `    ``int``[,] query) ` `{ ` `    ``// Precompute the results ` `    ``precompute(k); ` `  `  `    ``// Iterating through the queries ` `    ``for` `(``int` `i = 0; i < q; i++) { ` `   `  `        ``int` `l = query[i, 0], ` `            ``r = query[i, 1]; ` `   `  `        ``// Getting count of range ` `        ``// in range [L, R] ` `        ``int` `cnt = prefix[r] ` `                  ``- prefix[l - 1]; ` `        ``Console.Write(cnt +``"\n"``); ` `    ``} ` `} ` `   `  `// Driver code ` `public` `static` `void` `Main(String[] args) ` `{ ` `  `  `    ``int` `[,]query ` `        ``= { { 1, 11 }, ` `            ``{ 5, 15 }, ` `            ``{ 2, 24 } }; ` `    ``int` `k = 2, q = query.GetLength(0); ` `   `  `    ``performQueries(k, q, query); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992 `

Output:

```3
2
5
``` My Personal Notes arrow_drop_up
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