# Number of Co-prime pairs obtained from the sum of digits of elements in the given range

• Last Updated : 30 Jun, 2021

Given two numbers A and B where 1 <= A <= B. The task is to count the number of pairs whose elements are co-prime where pairs are formed from the sum of the digits of the elements in the given range.

Note: Two pairs are counted as distinct if at least one of the number in the pair is different. It may be assumed that the maximum digit sum can be 162.

Examples:

```Input: 12 15
Output: 4
12 = 1+2 = 3
13 = 1+3 = 4
14 = 1+4 = 5
15 = 1+5 = 6
Thus pairs who are co-prime to each other are
(3, 4), (3, 5), (4, 5), (5, 6)

Input: 7 10
Output: 6```

Method-1:

• Consider each and every element from a to b.
• Find the sum of the digits of every element and store it into a vector.
• Consider each and every pair one by one and check if the gcd of the elements of that pair is 1.
• If yes, count that pair as it is co-prime.
• Print the count of pairs that are co-prime.

Below is the implementation of the above approach:

## C++

 `// C++ program to count the pairs``// whose sum of digits is co-prime``#include ``using` `namespace` `std;` `// Function to find the elements``// after doing the sum of digits``int` `makePairs(vector<``int``> &pairs, ``int` `a, ``int` `b)``{``    ` `    ``// Traverse from a  to b``    ``for` `(``int` `i = a; i <= b; i++)``    ``{``        ` `    ``// Find the sum of the digits of the elements``    ``// in the given range one by one``      ``int` `sumOfDigits = 0, k = i;``      ``while``(k>0)``      ``{``          ``sumOfDigits += k%10;``          ``k /= 10;``      ``}``      ``if` `(sumOfDigits <= 162)``      ``pairs.push_back(sumOfDigits);``    ``}``}` `// Function to count the co-prime pairs``int` `countCoPrime(``int` `a, ``int` `b){``    ``vector<``int``> pairs;``    ` `    ``// Function to make the pairs``    ``// by doing the sum of digits``    ``makePairs(pairs, a, b);``    ``int` `count = 0;``    ` `    ``// Count pairs that are co-primes``    ``for``(``int` `i = 0; i < pairs.size(); i++)``       ``for` `(``int` `j = i+1; j < pairs.size(); j++)``          ``if` `(__gcd(pairs[i], pairs[j]) == 1)``                 ``count++;``             ` `   ``return` `count;``    ` `}` `// Driver code``int` `main()``{``    ``int` `a = 12, b = 15;` `    ``// Function to count the pairs``    ``cout << countCoPrime(a, b) ;` `    ``return` `0;``}`

## Java

 `// Java program to count the pairs``// whose sum of digits is co-prime``import` `java.util.*;` `class` `GFG``{``static` `int` `GCD(``int` `a, ``int` `b)``{``if` `(b == ``0``) ``return` `a;``return` `GCD(b, a % b);``}``// Function to find the elements``// after doing the sum of digits``static` `void` `makePairs(Vector pairs,``                      ``int` `a, ``int` `b)``{``    ` `    ``// Traverse from a to b``    ``for` `(``int` `i = a; i <= b; i++)``    ``{``        ` `    ``// Find the sum of the digits``    ``// of the elements in the given``    ``// range one by one``    ``int` `sumOfDigits = ``0``, k = i;``    ``while``(k > ``0``)``    ``{``        ``sumOfDigits += k % ``10``;``        ``k /= ``10``;``    ``}``    ``if` `(sumOfDigits <= ``162``)``        ``pairs.add(sumOfDigits);``    ``}``}` `// Function to count``// the co-prime pairs``static` `int` `countCoPrime(``int` `a, ``int` `b)``{``    ``Vector pairs = ``new` `Vector();``    ` `    ``// Function to make the pairs``    ``// by doing the sum of digits``    ``makePairs(pairs, a, b);``    ``int` `count = ``0``;``    ` `    ``// Count pairs that are co-primes``    ``for``(``int` `i = ``0``; i < pairs.size(); i++)``    ``for` `(``int` `j = i+``1``; j < pairs.size(); j++)``        ``if` `(GCD(pairs.get(i),``                ``pairs.get(j)) == ``1``)``                ``count++;``            ` `return` `count;``    ` `}` `// Driver code``public` `static` `void` `main(String args[])``{``    ``int` `a = ``12``, b = ``15``;` `    ``// Function to count the pairs``    ``System.out.println(countCoPrime(a, b));``}``}`  `// This code is contributed by Arnab Kundu`

## Python3

 `# Python3 program to count the pairs``# whose sum of digits is co-prime``from` `math ``import` `gcd``# Function to find the elements``# after doing the sum of digits``def` `makePairs(pairs, a, b):``    ``# Traverse from a to b``    ``for` `i ``in` `range``(a,b``+``1``,``1``):``        ``# Find the sum of the digits of the elements``        ``# in the given range one by one``        ``sumOfDigits ``=` `0``        ``k ``=` `i``        ``while``(k>``0``):``            ``sumOfDigits ``+``=` `k``%``10``            ``k ``=` `int``(k ``/` `10``)``    ` `        ``if` `(sumOfDigits <``=` `162``):``            ``pairs.append(sumOfDigits)``    ` `# Function to count the co-prime pairs``def` `countCoPrime(a, b):``    ``pairs ``=` `[]``    ` `    ``# Function to make the pairs``    ``# by doing the sum of digits``    ``makePairs(pairs, a, b)``    ``count ``=` `0``    ` `    ``# Count pairs that are co-primes``    ``for` `i ``in` `range``(``0``,``len``(pairs),``1``):``            ``for` `j ``in` `range``(i``+``1``,``len``(pairs),``1``):``                ``if` `(gcd(pairs[i], pairs[j]) ``=``=` `1``):``                    ``count ``+``=` `1``                    ` `    ``return` `count``    ` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``a ``=` `12``    ``b ``=` `15` `    ``# Function to count the pairs``    ``print` `(countCoPrime(a, b))` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# program to count the pairs``// whose sum of digits is co-prime``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{``public` `static` `int` `GCD(``int` `a, ``int` `b)``{``    ``if` `(b == 0)``    ``{``        ``return` `a;``    ``}``    ``return` `GCD(b, a % b);``}` `// Function to find the elements``// after doing the sum of digits``public` `static` `void` `makePairs(List<``int``> pairs,``                                ``int` `a, ``int` `b)``{` `    ``// Traverse from a to b``    ``for` `(``int` `i = a; i <= b; i++)``    ``{` `    ``// Find the sum of the digits``    ``// of the elements in the given``    ``// range one by one``    ``int` `sumOfDigits = 0, k = i;``    ``while` `(k > 0)``    ``{``        ``sumOfDigits += k % 10;``        ``k /= 10;``    ``}``    ``if` `(sumOfDigits <= 162)``    ``{``        ``pairs.Add(sumOfDigits);``    ``}``    ``}``}` `// Function to count``// the co-prime pairs``public` `static` `int` `countCoPrime(``int` `a, ``int` `b)``{``    ``List<``int``> pairs = ``new` `List<``int``>();` `    ``// Function to make the pairs``    ``// by doing the sum of digits``    ``makePairs(pairs, a, b);``    ``int` `count = 0;` `    ``// Count pairs that are co-primes``    ``for` `(``int` `i = 0;``             ``i < pairs.Count; i++)``    ``{``    ``for` `(``int` `j = i + 1;``             ``j < pairs.Count; j++)``    ``{``        ``if` `(GCD(pairs[i], pairs[j]) == 1)``        ``{``                ``count++;``        ``}``    ``}``    ``}` `    ``return` `count;` `}` `// Driver code``public` `static` `void` `Main(``string``[] args)``{``    ``int` `a = 12, b = 15;` `    ``// Function to count the pairs``    ``Console.WriteLine(countCoPrime(a, b));``}``}` `// This code is contributed``// by Shrikant13`

## Javascript

 ``

Output:

`4`

Method-2:
As mentioned in the question, the maximum sum can be 162. So, find out the frequency of numbers having their digit sum from 1 to 162 in range A to B and store the frequency in the array. Later, find the answer using this frequency.
Since,

Number, Frequency
1, 0
2, 0
3, 1
4, 1
5, 1
6, 1
7, 0
8, 0
., .
., .
162, 0
Thus Number of gcd pairs = freq(3)*freq(4) + freq(3)*freq(5) + freq(4)*freq(5) + freq(5)* freq(6)
= 1+1+1+1
= 4

Thus pairs who are co-prime to each other are (3,4), (3,5), (4,5), (5,6) i.e the answer is 4.

Below is the required implementation:

## C++

 `// C++ program to count the pairs``// whose sum of digits is co-prime``#include ``#define ll long long int``using` `namespace` `std;` `// Recursive function to return the frequency``// of numbers having their sum of digits i``ll recursive(ll idx, ll sum, ll tight, string st,``             ``ll dp[20][2][166], ll num)``{``    ``if` `(idx == num)``        ``// Returns 1 or 0``        ``return` `sum == 0;` `    ``// Returns value of the dp if already stored``    ``if` `(dp[idx][tight][sum] != -1)``        ``return` `dp[idx][tight][sum];` `    ``bool` `newTight;``    ``ll ans = 0;``    ``ll d;` `    ``// Loop from digit 0 to 9``    ``for` `(d = 0; d < 10; ++d) {``        ``newTight = ``false``;``        ``if` `(tight && st[idx] - ``'0'` `< d)``            ``continue``;` `        ``// To change the tight to 1``        ``if` `(tight && st[idx] - ``'0'` `== d)``            ``newTight = ``true``;` `        ``// Calling the recursive function to find the frequency``        ``if` `(sum >= d)``            ``ans += recursive(idx + 1, sum - d,``                             ``newTight, st, dp, num);``    ``}` `    ``return` `dp[idx][tight][sum] = ans;``}` `// Function to find out frequency of numbers``// from 1 to N having their sum of digits``// from 1 to 162 and store them in array``vector formArray(ll N)``{``    ``ll dp[20][2][166];``    ``memset``(dp, -1, ``sizeof` `dp);` `    ``// Number to string conversion``    ``ostringstream x;``    ``x << N;``    ``string st = x.str();``    ``ll num = st.size();` `    ``vector arr;``    ``for` `(``int` `i = 1; i <= 162; ++i) {` `        ``// Calling the recursive function``        ``// and pushing it into array``        ``arr.push_back(recursive(0, i, 1, st, dp, num));``    ``}` `    ``return` `arr;``}` `// Function to find the pairs``ll findPair(ll a, ll b)``{``    ``// Calling the  formArray function of a-1 numbers``    ``vector arr_smaller = formArray(a - 1);` `    ``// Calling the  formArray function of b numbers``    ``vector arr_greater = formArray(b);` `    ``// Subtracting the frequency of higher number array with lower``    ``// number array and thus finding the range of``    ``// numbers from a to b having sum from 1 to 162``    ``for` `(``int` `i = 0; i < arr_greater.size(); ++i)``        ``arr_greater[i] -= arr_smaller[i];` `    ``int` `ans = 0;``    ``for` `(``int` `i = 1; i <= 162; ++i) {``        ``for` `(``int` `j = i + 1; j <= 162; ++j) {` `            ``// To find out total number of pairs``            ``// which are co-prime``            ``if` `(__gcd(i, j) == 1)``                ``ans = (ans + arr_greater[i - 1] * arr_greater[j - 1]);``        ``}``    ``}` `    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``ll a = 12, b = 15;` `    ``// Function to count the pairs``    ``cout << findPair(a, b);` `    ``return` `0;``}`

## Java

 `// Java program to count the pairs``// whose sum of digits is co-prime``import` `java.io.*;``import` `java.util.*;``class` `GFG``{` `  ``static` `int` `gcd(``int` `a, ``int` `b)``  ``{``    ``if` `(b == ``0``)``      ``return` `a;``    ``return` `gcd(b, a % b);        ``  ``}` `  ``// Recursive function to return the frequency``  ``// of numbers having their sum of digits i``  ``static` `long` `recursive(``long` `idx, ``long` `sum, ``long` `tight,``                        ``String st, ``long``[][][] dp, ``long` `num)``  ``{``    ``if` `(idx == num)``    ``{` `      ``// Returns 1 or 0``      ``return` `sum == ``0` `? ``1` `: ``0``;``    ``}` `    ``// Returns value of the dp if already stored``    ``if` `(dp[(``int``)idx][(``int``)tight][(``int``)sum] != -``1``)``      ``return` `dp[(``int``)idx][(``int``)tight][(``int``)sum];` `    ``long` `newTight;``    ``long` `ans = ``0``;``    ``long` `d;` `    ``// Loop from digit 0 to 9``    ``for` `(d = ``0``; d < ``10``; ++d)``    ``{``      ``newTight = ``0``;``      ``if` `(tight == ``1` `&& st.charAt((``int``)idx) - ``'0'` `< d)``        ``continue``;` `      ``// To change the tight to 1``      ``if` `(tight == ``1` `&& st.charAt((``int``)idx) - ``'0'` `== d)``        ``newTight = ``1``;` `      ``// Calling the recursive function to find the frequency``      ``if` `(sum >= d)``        ``ans += recursive(idx + ``1``, sum - d,``                         ``(``int``)newTight, st, dp, num);``    ``}``    ``return` `dp[(``int``)idx][(``int``)tight][(``int``)sum] = ans;``  ``}` `  ``// Function to find out frequency of numbers``  ``// from 1 to N having their sum of digits``  ``// from 1 to 162 and store them in array``  ``static` `ArrayList formArray(``long` `N)``  ``{``    ``long``[][][] dp = ``new` `long``[``20``][``2``][``166``];``    ``for` `(``long``[][] innerRow: dp)``    ``{``      ``for` `(``long``[] innerInnerRow: innerRow)``      ``{``        ``Arrays.fill(innerInnerRow, -``1``);``      ``}``    ``}` `    ``// Number to string conversion``    ``String st = String.valueOf(N);``    ``long` `num = st.length();` `    ``ArrayList arr = ``new` `ArrayList();``    ``for` `(``int` `i = ``1``; i <= ``162``; ++i)``    ``{``      ``// Calling the recursive function``      ``// and pushing it into array``      ``arr.add(recursive(``0``, i, ``1``, st, dp, num));``    ``}``    ``return` `arr;``  ``}` `  ``// Function to find the pairs``  ``static` `long` `findPair(``long` `a, ``long` `b)``  ``{` `    ``// Calling the  formArray function of a-1 numbers``    ``ArrayList arr_smaller = formArray(a - ``1``);` `    ``// Calling the  formArray function of b numbers``    ``ArrayList arr_greater = formArray(b);` `    ``// Subtracting the frequency of higher number array with lower``    ``// number array and thus finding the range of``    ``// numbers from a to b having sum from 1 to 162``    ``for` `(``int` `i = ``0``; i < arr_greater.size(); ++i)``    ``{``      ``arr_greater.set(i,arr_greater.get(i)-arr_smaller.get(i));``    ``}``    ``long` `ans = ``0``;``    ``for` `(``int` `i = ``1``; i <= ``162``; ++i)``    ``{``      ``for` `(``int` `j = i + ``1``; j <= ``162``; ++j)``      ``{``        ``// To find out total number of pairs``        ``// which are co-prime``        ``if` `(gcd(i, j) == ``1``)``        ``{``          ``ans = (ans + arr_greater.get(i-``1``) * arr_greater.get(j-``1``));``        ``}``      ``}``    ``}``    ``return` `ans;``  ``}` `  ``// Driver code``  ``public` `static` `void` `main (String[] args)``  ``{``    ``long` `a = ``12``, b = ``15``;` `    ``// Function to count the pairs``    ``System.out.println(findPair(a, b));``  ``}``}` `// This code is contributed by avanitrachhadiya2155`

## Python3

 `# Python3 program to count``# the pairs whose sum of``# digits is co-prime``import` `math` `# Recursive function to``# return the frequency of``# numbers having their sum``# of digits i``def` `recursive(idx, ``sum``,``              ``tight, st,``              ``dp, num):` `    ``if` `(idx ``=``=` `num):``      ` `        ``# Returns 1 or 0``        ``return` `sum` `=``=` `0` `    ``# Returns value of the dp``    ``# if already stored``    ``if` `(dp[idx][tight][``sum``] !``=` `-``1``):``        ``return` `dp[idx][tight][``sum``]` `    ``ans ``=` `0` `    ``# Loop from digit 0 to 9``    ``for` `d ``in` `range``(``10``):``        ``newTight ``=` `False``        ``if` `(tight ``and` `ord``(st[idx]) ``-``            ``ord``(``'0'``) < d):``            ``continue` `        ``# To change the tight to 1``        ``if` `(tight ``and` `ord``(st[idx]) ``-``            ``ord``(``'0'``) ``=``=` `d):``            ``newTight ``=` `True` `        ``# Calling the recursive``        ``# function to find the``        ``# frequency``        ``if` `(``sum` `>``=` `d):``            ``ans ``+``=` `recursive(idx ``+` `1``,``                             ``sum` `-` `d,``                             ``newTight,``                             ``st, dp, num)` `    ``dp[idx][tight][``sum``] ``=` `ans``    ``return` `dp[idx][tight][``sum``]` `# Function to find out frequency``# of numbers from 1 to N having``# their sum of digits from 1 to``# 162 and store them in array``def` `formArray(N):` `    ``dp ``=` `[[[``-``1` `for` `x ``in` `range``(``166``)]``               ``for` `y ``in` `range``(``2``)]``               ``for` `z ``in` `range``(``20``)]` `    ``# Number to string conversion``    ``st ``=` `str``(N)``    ``num ``=` `len``(st)` `    ``arr ``=` `[]``    ``for` `i ``in` `range``(``1``, ``163``):` `        ``# Calling the recursive function``        ``# and pushing it into array``        ``arr.append(recursive(``0``, i, ``1``,``                             ``st, dp, num))``    ``return` `arr` `# Function to find the pairs``def` `findPair(a, b):` `    ``# Calling the  formArray``    ``# function of a-1 numbers``    ``arr_smaller ``=` `formArray(a ``-` `1``)` `    ``# Calling the  formArray``    ``# function of b numbers``    ``arr_greater ``=` `formArray(b)` `    ``# Subtracting the frequency of``    ``# higher number array with lower``    ``# number array and thus finding``    ``# the range of numbers from a to``    ``# b having sum from 1 to 162``    ``for` `i ``in` `range``(``len``(arr_greater)):``        ``arr_greater[i] ``-``=` `arr_smaller[i]` `    ``ans ``=` `0``    ``for` `i ``in` `range``(``1``, ``163``):``        ``for` `j ``in` `range``(i ``+` `1``, ``163``):` `            ``# To find out total number``            ``# of pairs which are co-prime``            ``if` `(math.gcd(i, j) ``=``=` `1``):``                ``ans ``=` `(ans ``+` `arr_greater[i ``-` `1``] ``*``                       ``arr_greater[j ``-` `1``])``    ``return` `ans` `# Driver code``if` `__name__ ``=``=` `"__main__"``:``    ``a ``=` `12``    ``b ``=` `15` `    ``# Function to count the pairs``    ``print``(findPair(a, b))` `# This code is contributed by Chitranayal`

## C#

 `// C# program to count the pairs``// whose sum of digits is co-prime``using` `System;``using` `System.Collections.Generic;` `class` `GFG{``    ` `static` `int` `gcd(``int` `a, ``int` `b)``{``    ``if` `(b == 0)``        ``return` `a;``    ``return` `gcd(b, a % b);        ``}`` ` `// Recursive function to return the frequency``// of numbers having their sum of digits i``static` `long` `recursive(``long` `idx, ``long` `sum, ``long` `tight,``                      ``string` `st, ``long``[,,] dp, ``long` `num)``{``    ``if` `(idx == num)``    ``{``        ` `        ``// Returns 1 or 0``        ``return` `sum == 0 ? 1 : 0;``    ``}``    ` `    ``// Returns value of the dp if already stored``    ``if` `(dp[(``int``)idx, (``int``)tight, (``int``)sum] != -1)``        ``return` `dp[(``int``)idx, (``int``)tight, (``int``)sum];``    ` `    ``long` `newTight;``    ``long` `ans = 0;``    ``long` `d;``    ` `    ``// Loop from digit 0 to 9``    ``for` `(d = 0; d < 10; ++d)``    ``{``        ``newTight = 0;``        ` `        ``if` `(tight == 1 && st[((``int``)idx)] - ``'0'` `< d)``            ``continue``;``        ` `        ``// To change the tight to 1``        ``if` `(tight == 1 && st[((``int``)idx)] - ``'0'` `== d)``            ``newTight = 1;``        ` `        ``// Calling the recursive function to``        ``// find the frequency``        ``if` `(sum >= d)``            ``ans += recursive(idx + 1, sum - d,``                   ``(``int``)newTight, st, dp, num);``    ``}``    ``return` `dp[(``int``)idx, (``int``)tight, (``int``)sum] = ans;``}`` ` `// Function to find out frequency of numbers``// from 1 to N having their sum of digits``// from 1 to 162 and store them in array``static` `List<``long``> formArray(``long` `N)``{``    ``long``[,,] dp = ``new` `long``[20, 2, 166];``    ``for``(``int` `i = 0; i < 20; i++)``    ``{``        ``for``(``int` `j = 0; j < 2; j++)``        ``{``            ``for``(``int` `k = 0; k < 166; k++)``            ``{``                ``dp[i, j, k] = -1;``            ``}``        ``}``    ``}``    ` `    ``// Number to string conversion``    ``string` `st = N.ToString();``    ``long` `num = st.Length;``    ` `    ``List<``long``> arr = ``new` `List<``long``>();``    ``for` `(``int` `i = 1; i <= 162; ++i)``    ``{``        ` `        ``// Calling the recursive function``        ``// and pushing it into array``        ``arr.Add(recursive(0, i, 1, st, dp, num));``    ``}``    ``return` `arr;``}` `// Function to find the pairs``static` `long` `findPair(``long` `a, ``long` `b)``{` `    ``// Calling the  formArray function of a-1 numbers``    ``List<``long``> arr_smaller = formArray(a - 1);``    ` `    ``// Calling the  formArray function of b numbers``    ``List<``long``> arr_greater = formArray(b);``    ` `    ``// Subtracting the frequency of higher number``    ``// array with lower number array and thus``    ``// finding the range of numbers from a to b``    ``// having sum from 1 to 162``    ``for``(``int` `i = 0; i < arr_greater.Count; ++i)``    ``{``        ``arr_greater[i] = arr_greater[i] -``                         ``arr_smaller[i];``    ``}``    ``long` `ans = 0;``    ` `    ``for``(``int` `i = 1; i <= 162; ++i)``    ``{``        ``for``(``int` `j = i + 1; j <= 162; ++j)``        ``{``            ` `            ``// To find out total number of pairs``            ``// which are co-prime``            ``if` `(gcd(i, j) == 1)``            ``{``                ``ans = (ans + arr_greater[i - 1] *``                             ``arr_greater[j - 1]);``            ``}``        ``}``    ``}``    ``return` `ans;``}` `// Driver code``static` `public` `void` `Main()``{``    ``long` `a = 12, b = 15;``    ` `    ``// Function to count the pairs``    ``Console.WriteLine(findPair(a, b));``}``}` `// This code is contributed by rag2127`

## Javascript

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Output:

`4`

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