# Extended Midy's theorem in C++

C++Server Side ProgrammingProgramming

Midy’s Theorem is a statement used for decimal expansion of numbers denoted by n/p, where n is any number and p is a prime number and a/p has a repeating decimal with even period.

In Extended Midy’s Theorem, the repeating portion is divided into m digits, then their sum is a multiple of 10m - 1.

Program to illustrate Extended Midy’s Theorem:

## Example

Live Demo

#include <bits/stdc++.h>
using namespace std;

string findDecimalValue(int num, int den) {

string res;
unordered_map<int, int> mp;
int rem = num % den;

while ((rem != 0) && (mp.find(rem) == mp.end())) {

mp[rem] = res.length();
rem = rem * 10;
int part = rem / den;
res += to_string(part);
rem = rem % den;
}
return (rem == 0) ? "-1" : res.substr(mp[rem]);
}

bool isPrime(int n) {

for (int i = 2; i <= n / 2; i++)
if (n % i == 0)
return false;
return true;
}

void ExtendedMidysAlgo(string str, int n, int m) {

if (!isPrime(n)) {
cout<<"Denominator is not prime, thus Extended Midy's theorem is not applicable";
return;
}

int l = str.length();
int part1 = 0, part2 = 0;
if (l % 2 == 0 && l % m == 0) {

int part[m] = { 0 }, sum = 0, res = 0;
for (int i = 0; i < l; i++) {
int var = i / m;
part[var] = part[var] * 10 + (str[i] - '0');
}
for (int i = 0; i < m; i++) {
sum = sum + part[i];
cout << part[i] << " ";
}
cout << endl;
res = pow(10, m) - 1;
if (sum % res == 0)
cout << "Extended Midy's theorem holds!";
else
cout << "Extended Midy's theorem doesn't hold!";
}
else if (l % 2 != 0) {
cout << "The repeating decimal is of odd length thus Extended Midy's theorem is not applicable";
}
else if (l % m != 0) {
cout<<" The repeating decimal can not be divided into m digits";
}
}

// Driver code
int main()
{
int numr = 1, denr = 17, m = 4;
string res = findDecimalValue(numr, denr);
if (res == "-1")
cout << "The fraction does not have repeating decimal";
else {
cout << "Repeating decimal = " << res << endl;
ExtendedMidysAlgo(res, denr, m);
}
return 0;
}

## Output −

Repeating decimal = 0588235294117647
588 2352 9411 7647
Extended Midy's theorem holds!