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Smallest root of the equation x^2 + s(x)*x – n = 0, where s(x) is the sum of digits of root x.
  • Last Updated : 22 Apr, 2021

You are given an integer n, find the smallest positive integer root of equation x, or else print -1 if no roots are found.
Equation: x^2 + s(x)*x – n = 0
where x, n are positive integers, s(x) is the function, equal to the sum of digits of number x in the decimal number system. 
1 <= N <= 10^18

Examples: 

Input: N = 110 
Output: 10 
Explanation: x = 10 is the minimum root. 
             As s(10) = 1 + 0 = 1 and 
             102 + 1*10 - 110=0.  

Input: N = 4
Output: -1 
Explanation: there are no roots of the 
equation possible

A naive approach will be to iterate through all the possible values of X and find out if any such root exists but this won’t be possible as the value of n is very large.

An efficient approach will be as follows 
Firstly let’s find the interval of possible values of s(x). Hence x^2 <= N and N <= 10^18, x <= 109. In other words, for every considerable solution x the decimal length of x does not extend 10 digits. So Smax = s(9999999999) = 10*9 = 90. 
Let’s brute force the value of s(x) (0 <= s(x) <= 90). Now we have an ordinary square equation. The deal is to solve the equation and to check that the current brute forced value of s(x) is equal to sum of digits of the solution. If the solution exists and the equality holds, we should get the answer and store the minimum of the roots possible.

Below is the implementation of the above approach  



C++




// CPP program to find smallest value of root
// of an equation under given constraints.
#include <bits/stdc++.h>
using namespace std;
 
// function to check if the sum of digits is
// equal to the summation assumed
bool check(long long a, long long b)
{
    long long int c = 0;
 
    // calculate the sum of digit
    while (a != 0) {
        c = c + a % 10;
        a = a / 10;
    }
 
    return (c == b);
}
 
// function to find the largest root possible.
long long root(long long n)
{
    bool found = 0;
    long long mx = 1e18;
 
    // iterate for all possible sum of digits.
    for (long long i = 0; i <= 90; i++) {
 
        // check if discriminent is a perfect square.
        long long s = i * i + 4 * n;
        long long sq = sqrt(s);
 
        // check if discriminent is a perfect square and
        // if it as perefect root of the equation
        if (sq * sq == s && check((sq - i) / 2, i)) {
            found = 1;
            mx = min(mx, (sq - i) / 2);
        }
    }
 
    // function returns answer
    if (found)
        return mx;
    else
        return -1;
}
 
// driver program to check the above function
int main()
{
    long long n = 110;
    cout << root(n);
}

Java




// Java program to find smallest value of root
// of an equation under given constraints.
 
class GFG{
// function to check if the sum of digits is
// equal to the summation assumed
static boolean check(long a, long b)
{
    long c = 0;
 
    // calculate the sum of digit
    while (a != 0) {
        c = c + a % 10;
        a = a / 10;
    }
 
    return (c == b);
}
 
// function to find the largest root possible.
static long root(long n)
{
    boolean found = false;
    long mx = (long)1E18;
 
    // iterate for all possible sum of digits.
    for (long i = 0; i <= 90; i++) {
 
        // check if discriminent is a perfect square.
        long s = i * i + 4 * n;
        long sq = (long)Math.sqrt(s);
 
        // check if discriminent is a perfect square and
        // if it as perefect root of the equation
        if (sq * sq == s && check((sq - i) / 2, i)) {
            found = true;
            mx = Math.min(mx, (sq - i) / 2);
        }
    }
 
    // function returns answer
    if (found)
        return mx;
    else
        return -1;
}
 
// driver program to check the above function
public static void main(String[] args)
{
    long n = 110;
    System.out.println(root(n));
}
}
// This code is contributed by mits

Python3




# Python3 program to find smallest
# value of root of an equation
# under given constraints.
import math
 
# function to check if the sum
# of digits is equal to the
# summation assumed
def check(a, b):
    c = 0;
 
    # calculate the
    # sum of digit
    while (a != 0):
        c = c + a % 10;
        a = int(a / 10);
 
    return True if(c == b) else False;
 
# function to find the
# largest root possible.
def root(n):
    found = False;
     
    # float(1E+18)
    mx = 1000000000000000001;
 
    # iterate for all
    # possible sum of digits.
    for i in range(91):
 
        # check if discriminent
        # is a perfect square.
        s = i * i + 4 * n;
        sq = int(math.sqrt(s));
 
        # check if discriminent is
        # a perfect square and
        # if it as perefect root
        # of the equation
        if (sq * sq == s and
            check(int((sq - i) / 2), i)):
            found = True;
            mx = min(mx, int((sq-i) / 2));
 
    # function returns answer
    if (found):
        return mx;
    else:
        return -1;
 
# Driver Code
n = 110;
print(root(n));
 
# This code is contributed by mits

C#




//C# program to find smallest value of root
// of an equation under given constraints.
using System;
public class GFG{
    // function to check if the sum of digits is
    // equal to the summation assumed
    static bool check(long a, long b)
    {
        long c = 0;
 
        // calculate the sum of digit
        while (a != 0) {
            c = c + a % 10;
            a = a / 10;
        }
 
        return (c == b);
    }
 
    // function to find the largest root possible.
    static long root(long n)
    {
        bool found = false;
        long mx = (long)1E18;
 
        // iterate for all possible sum of digits.
        for (long i = 0; i <= 90; i++) {
 
            // check if discriminent is a perfect square.
            long s = i * i + 4 * n;
            long sq = (long)Math.Sqrt(s);
 
            // check if discriminent is a perfect square and
            // if it as perefect root of the equation
            if (sq * sq == s && check((sq - i) / 2, i)) {
                found = true;
                mx = Math.Min(mx, (sq - i) / 2);
            }
        }
 
        // function returns answer
        if (found)
            return mx;
        else
            return -1;
    }
 
    // driver program to check the above function
    public static void Main()
    {
        long n = 110;
        Console.Write(root(n));
    }
}
// This code is contributed by Raput-Ji

PHP




<?php
// PHP program to find
// smallest value of root
// of an equation under
// given constraints.
 
// function to check if
// the sum of digits is
// equal to the summation
// assumed
function check($a, $b)
{
    $c = 0;
 
    // calculate the
    // sum of digit
    while ($a != 0)
    {
        $c = $c + $a % 10;
        $a = (int)($a / 10);
    }
 
    return ($c == $b) ? true : false;
}
 
// function to find the
// largest root possible.
function root($n)
{
    $found = false;
     
    // float(1E+18)
    $mx = 1000000000000000001;
 
    // iterate for all
    // possible sum of digits.
    for ($i = 0; $i <= 90; $i++)
    {
 
        // check if discriminent
        // is a perfect square.
        $s = $i * $i + 4 * $n;
        $sq = (int)(sqrt($s));
 
        // check if discriminent is
        // a perfect square and
        // if it as perefect root
        // of the equation
        if ($sq * $sq == $s &&
            check((int)(($sq - $i) / 2), $i))
        {
            $found = true;
            $mx = min($mx, (int)(($sq -
                                  $i) / 2));
        }
    }
 
    // function returns answer
    if ($found)
        return $mx;
    else
        return -1;
}
 
// Driver Code
$n = 110;
echo root($n);
 
// This code is contributed by mits
?>

Javascript




<script>
 
// Javascript program to find smallest
// value of root of an equation under
// given constraints.   
 
// Function to check if the sum of digits is
// equal to the summation assumed
function check(a, b)
{
    var c = 0;
 
    // Calculate the sum of digit
    while (a != 0)
    {
        c = c + a % 10;
        a = parseInt(a / 10);
    }
 
    return (c == b);
}
 
// Function to find the largest root possible.
function root(n)
{
    var found = false;
    var mx =  1E18;
 
    // Iterate for all possible
    // sum of digits.
    for(i = 0; i <= 90; i++)
    {
         
        // Check if discriminent is a
        // perfect square.
        var s = i * i + 4 * n;
        var sq =  Math.sqrt(s);
 
        // Check if discriminent is a
        // perfect square and if it as
        // perefect root of the equation
        if (sq * sq == s && check((sq - i) / 2, i))
        {
            found = true;
            mx = Math.min(mx, (sq - i) / 2);
        }
    }
 
    // Function returns answer
    if (found)
        return mx;
    else
        return -1;
}
 
// Driver code
var n = 110;
 
document.write(root(n));
 
// This code is contributed by todaysgaurav
 
</script>

Output: 

10 

 

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