# Midy’s theorem in C++

We are given integer values as a_num that will store the numerator and p_den that will store the denominator which should be a prime number. The task is to check whether the operations performed on a_num after dividing with p_den proves the midy’s theorem or not.

## Steps to prove Midy’s theorem are-

• Input numerator as a_num and denominator as p_den which should always be a prime value.

• Divide the numbers. Check for the repeating decimal values.

• Store the decimal values until they are not repeating.

• Check whether the digits are even, if yes, then break them into halves

• Add both the numbers. If the output is a string of 9 then it proves Midy’s theorem.

Let us see various input output scenarios for this -

In − int a_num = 1 and int p_den = 19

Out − Repeating decimals are: 052631578947368421 Proved Midy's theorem

Explanation − Follow the steps mentioned above to check for Midy’s theorem i.e.

• Divide 1 / 19 = 052631578947368421

• Repeating decimal values are-: 052631578947368421.

• Divide the digits into halves i.e. 052631578 947368421.

• Add both the halves i.e. 052631578 + 947368421 = 999,999,999.

• As we can see, 999,999,999 is the string of 9 which proves the Midy’s theorem.

In  −int a_num = 49, int p_den = 7

Out − No Repeating Decimal

Explanation − As we can see 49/7 generates no decimal value since 49 is completely divisible by 7. So, the output is No Repeating Decimal.

## Approach used in the below program is as follows

• Input to integer values as int a_num and int p_den.

• Call function as Midys_theorem(a_num, p_den) to prove Midy's theorem.

• Inside the function check_Midys()

• Create variables as int first to 0 and int last to 0

• Check IF the function check(val) returns FALSE then print Midy’s theorem not applicable.

• ELSE IF len % 2 = 0 then start loop FOR from i to 0 till i less than len/2 and set first to first * 10 + (str[i] - '0') and last to last * 10 + (str[len / 2 + i] - '0') and print proved Midy’s theorem.

• ELSE, print Midy’s theorem not applicable.

• Inside the function Midys_theorem(int a_num, int p_den)

• Create a map type variable to map integer type values as map_val and clear the map.

• Set reminder as a_num % p_den.

• Start while reminder not equals to 0 AND map_val.find(reminder) equals to map_val.end() then set map_val[reminder] to result.length(), reminder to reminder * 10, temp to reminder / p_den, result to result + to_string(temp) and reminder to reminder % p_den.

• Check IF remainder = 0 then return -1 ELSE, set count to result.substr(map_val[reminder])

• Return count

• Inside the function bool check(int val)

• Start loop FOR from i to 2 till i less than val/2. Check IF val % i = 0 then return FALSE otherwise return TRUE.

## Example

#include <bits/stdc++.h>
using namespace std;
bool check(int val){
for(int i = 2; i <= val / 2; i++){
if(val % i == 0){
return false;
}
}
return true;
}
void check_Midys(string str, int val){
int len = str.length();
int first = 0;
int last = 0;

if(!check(val)){
cout<<"\nNot applicable for Midy's theorem";
}
else if(len % 2 == 0){
for(int i = 0; i < len / 2; i++){
first = first * 10 + (str[i] - '0');
last = last * 10 + (str[len / 2 + i] - '0');
}
cout<<"\nProved Midy's theorem";
}
else{
cout<<"\nNot applicable for Midy's theorem";
}
}
string Midys_theorem(int a_num, int p_den){
string result;
map<int, int> map_val;
map_val.clear();

int reminder = a_num % p_den;

while((reminder != 0) && (map_val.find(reminder) == map_val.end())){
map_val[reminder] = result.length();
reminder = reminder * 10;
int temp = reminder / p_den;
result += to_string(temp);
reminder = reminder % p_den;
}
if(reminder == 0){
return "-1";
}
else{
string count = result.substr(map_val[reminder]);
return count;
}
}
int main(){
int a_num = 1;
int p_den = 19;
string result = Midys_theorem(a_num, p_den);
if(result == "-1"){
cout<<"No Repeating Decimal";
}
else{
cout<<"Repeating decimals are: "<<result;
check_Midys(result, p_den);
}
return 0;
}

## Output

If we run the above code it will generate the following Output

Repeating decimals are: 052631578947368421
Proved Midy's theorem