Skip to content
Related Articles

Related Articles

Extended Midy’s theorem

View Discussion
Improve Article
Save Article
  • Last Updated : 03 Aug, 2022

According to Midy’s theorem, if the period of a repeating decimal for a / p          , where p is prime and a / p          is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example 1/7 = 0.14285714285.. is a repeating decimal with 142857 being repeated. Now, according to the theorem, it has even number of repeating digits i.e. 142857. Further if we divide this into two halves, we get 142 and 857. Thus, on adding these two, we get 999 which is string of 9s and matches our theorem. 

In Extended Midy’s theorem if we divide the repeating portion of a/p into m digits, then their sum is a multiple of 10m -1.

Suppose a = 1 and p = 17, 
a/p = 1/17 = 0.0588235294117647… 
So, 0588235294117647 is the repeating portion of the decimal expansion of 1/17. Its repeating portion has 16 digits and it can be divided into m digits where m can be 2, 4, 8. 
If we consider m = 4 then 0588235294117647 can be divided into 16/4 = 4 numbers and if we add these 4 numbers then result should be a multiple of 104 – 1 = 9999 i.e, 
0588 + 2352 + 9411 + 7647 = 19998 = 2*9999

C++




// C++ program to demonstrate extended
// Midy's theorem
#include <bits/stdc++.h>
using namespace std;
 
// Returns repeating sequence of a fraction.
// If repeating sequence doesn't exits,
// then returns -1
string fractionToDecimal(int numerator,
                         int denominator)
{
    string res;
 
    /* Create a map to store already seen remainders
    remainder is used as key and its position in
    result is stored as value. Note that we need
    position for cases like 1/6. In this case,
    the recurring sequence doesn't start from first
    remainder. */
    unordered_map<int, int> mp;
 
    // Find first remainder
    int rem = numerator % denominator;
 
    // Keep finding remainder until either remainder
    // becomes 0 or repeats
    while ((rem != 0) && (mp.find(rem) == mp.end())) {
 
        // Store this remainder
        mp[rem] = res.length();
 
        // Multiply remainder with 10
        rem = rem * 10;
 
        // Append rem / denr to result
        int res_part = rem / denominator;
        res += to_string(res_part);
 
        // Update remainder
        rem = rem % denominator;
    }
 
    return (rem == 0) ? "-1" : res.substr(mp[rem]);
}
 
// Checks whether a number is prime or not
bool isPrime(int n)
{
    for (int i = 2; i <= n / 2; i++)
        if (n % i == 0)
            return false;
    return true;
}
 
// If all conditions are met,
// it proves Extended Midy's theorem
void ExtendedMidys(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) {
 
        // Dividing repeated part into m parts
        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');
        }
 
        // Computing sum of parts.
        for (int i = 0; i < m; i++) {
            sum = sum + part[i];
            cout << part[i] << " ";
        }
         
        // Checking for Extended Midy
        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 = fractionToDecimal(numr, denr);
    if (res == "-1")
        cout << "The fraction does not"
             << " have repeating decimal";
    else {
        cout << "Repeating decimal = " << res << endl;
        ExtendedMidys(res, denr, m);
    }
    return 0;
}

Java




// Java program to demonstrate extended
// Midy's theorem
import java.util.*;
 
class GFG{
 
// Returns repeating sequence of a fraction.
// If repeating sequence doesn't exits,
// then returns -1
static String fractionToDecimal(int numerator,
                                int denominator)
{
    String res = "";
 
    /* Create a map to store already seen remainders
    remainder is used as key and its position in
    result is stored as value. Note that we need
    position for cases like 1/6. In this case,
    the recurring sequence doesn't start from first
    remainder. */
    HashMap<Integer, Integer> mp = new HashMap<>();
 
    // Find first remainder
    int rem = numerator % denominator;
 
    // Keep finding remainder until either remainder
    // becomes 0 or repeats
    while ((rem != 0) && !mp.containsKey(rem))
    {
         
        // Store this remainder
        mp.put(rem, res.length());
 
        // Multiply remainder with 10
        rem = rem * 10;
 
        // Append rem / denr to result
        int res_part = rem / denominator;
        res += res_part + "";
 
        // Update remainder
        rem = rem % denominator;
    }
 
    return (rem == 0) ? "-1" : res.substring(mp.get(rem));
}
 
// Checks whether a number is prime or not
static boolean isPrime(int n)
{
    for(int i = 2; i <= n / 2; i++)
        if (n % i == 0)
            return false;
             
    return true;
}
 
// If all conditions are met,
// it proves Extended Midy's theorem
static void ExtendedMidys(String str, int n, int m)
{
    if (!isPrime(n))
    {
        System.out.print("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)
    {
         
        // Dividing repeated part into m parts
        int []part = new int[m];
        int sum = 0, res = 0;
        for(int i = 0; i < l; i++)
        {
            int var = i / m;
            part[var] = part[var] * 10 +
                   (str.charAt(i) - '0');
        }
 
        // Computing sum of parts.
        for(int i = 0; i < m; i++)
        {
            sum = sum + part[i];
            System.out.print(part[i] + " ");
        }
         
        // Checking for Extended Midy
        System.out.println();
        res = (int)Math.pow(10, m) - 1;
         
        if (sum % res == 0)
            System.out.print("Extended Midy's " +
                             "theorem holds!");       
        else
            System.out.print("Extended Midy's " +
                             "theorem doesn't hold!");       
    }
    else if (l % 2 != 0)
    {
        System.out.print("The repeating decimal is of " +
                         "odd length thus Extended Midy's " +
                         "theorem is not applicable");
    }
    else if (l % m != 0)
    {
        System.out.print("The repeating decimal can " +
                         "not be divided into m digits");
    }
}
 
// Driver code
public static void main(String []args)
{
    int numr = 1, denr = 17, m = 4;
    String res = fractionToDecimal(numr, denr);
     
    if (res == "-1")
        System.out.print("The fraction does not " +
                         "have repeating decimal");
    else
    {
        System.out.println("Repeating decimal = " + res);
        ExtendedMidys(res, denr, m);
    }
}
}
 
// This code is contributed by rutvik_56

Python3




# Python3 program to demonstrate extended
# Midy's theorem
 
# Returns repeating sequence of a fraction.
# If repeating sequence doesn't exits,
# then returns -1
def fractionToDecimal(numerator, denominator):
    res = "";
 
    ''' Create a map to store already seen remainders
    remainder is used as key and its position in
    result is stored as value. Note that we need
    position for cases like 1/6. In this case,
    the recurring sequence doesn't start from first
    remainder. '''
    mp = dict();
 
    # Find first remainder
    rem = numerator % denominator;
 
    # Keep finding remainder until either remainder
    # becomes 0 or repeats
    while ((rem != 0) and (rem not in mp)):
 
        # Store this remainder
        mp[rem] = len(res);
 
        # Multiply remainder with 10
        rem = rem * 10;
 
        # Append rem / denr to result
        res_part = rem // denominator;
        res += str(res_part);
 
        # Update remainder
        rem = rem % denominator;
     
    if rem == 0:
        return "-1"
    return res[mp[rem]:]
     
 
# Checks whether a number is prime or not
def isPrime(n):
    for i in range(2, n // 2 + 1):
        if (n % i == 0):
            return False;
    return True;
 
 
# If all conditions are met,
# it proves Extended Midy's theorem
def ExtendedMidys(str, n, m):
    if (not isPrime(n)):
        print("Denominator is not prime, thus Extended Midy's theorem is not applicable");
        return;
     
 
    l = len(str);
    part1 = 0
    part2 = 0;
    if (l % 2 == 0 and l % m == 0):
 
        # Dividing repeated part into m parts
        part = [0 for _ in range(m)];
        sum = 0
        res = 0;
        for i in range(l):
            var = i // m
            part[var] = part[var] * 10 + int(str[i]);
         
 
        # Computing sum of parts.
        for i in range(m):
            sum = sum + part[i];
            print(part[i], end =  " ");
         
 
        # Checking for Extended Midy
        print()
        res = pow(10, m) - 1;
        if (sum % res == 0):
            print("Extended Midy's theorem holds!");
        else:
            print("Extended Midy's theorem doesn't hold!");
     
    elif (l % 2 != 0):
        print("The repeating decimal is of odd length thus Extended Midy's theorem is not applicable");
     
    elif (l % m != 0):
        print("The repeating decimal can not be divided into m digits");
     
 
 
# Driver code
numr = 1
denr = 17
m = 4;
res = fractionToDecimal(numr, denr);
if (res == "-1"):
    print("The fraction does not have repeating decimal");
else:
    print("Repeating decimal =", res);
    ExtendedMidys(res, denr, m);
 
 
 
# This code is contributed by phasing17

C#




// C# program to demonstrate extended
// Midy's theorem
using System;
using System.Collections;
using System.Collections.Generic;
 
class GFG{
 
// Returns repeating sequence of a fraction.
// If repeating sequence doesn't exits,
// then returns -1
static string fractionToDecimal(int numerator,
                                int denominator)
{
    string res = "";
 
    /* Create a map to store already seen remainders
    remainder is used as key and its position in
    result is stored as value. Note that we need
    position for cases like 1/6. In this case,
    the recurring sequence doesn't start from first
    remainder. */
    Dictionary<int,int> mp = new Dictionary<int,int>();
 
    // Find first remainder
    int rem = numerator % denominator;
 
    // Keep finding remainder until either remainder
    // becomes 0 or repeats
    while ((rem != 0) && !mp.ContainsKey(rem))
    {
         
        // Store this remainder
        mp[rem]= res.Length;
 
        // Multiply remainder with 10
        rem = rem * 10;
 
        // Append rem / denr to result
        int res_part = rem / denominator;
        res += res_part + "";
 
        // Update remainder
        rem = rem % denominator;
    }
 
    return (rem == 0) ? "-1" : res.Substring(mp[rem]);
}
 
// Checks whether a number is prime or not
static bool isPrime(int n)
{
    for(int i = 2; i <= n / 2; i++)
        if (n % i == 0)
            return false;
             
    return true;
}
 
// If all conditions are met,
// it proves Extended Midy's theorem
static void ExtendedMidys(string str, int n, int m)
{
    if (!isPrime(n))
    {
        Console.Write("Denominator is not prime, " +
                         "thus Extended Midy's theorem " +
                         "is not applicable");
        return;   
    }
 
    int l = str.Length;
     
    if (l % 2 == 0 && l % m == 0)
    {
         
        // Dividing repeated part into m parts
        int []part = new int[m];
        int sum = 0, res = 0;
        for(int i = 0; i < l; i++)
        {
            int var = i / m;
            part[var] = part[var] * 10 +
                   (str[i] - '0');
        }
 
        // Computing sum of parts.
        for(int i = 0; i < m; i++)
        {
            sum = sum + part[i];
            Console.Write(part[i] + " ");
        }
         
        // Checking for Extended Midy
        Console.WriteLine();
        res = (int)Math.Pow(10, m) - 1;
         
        if (sum % res == 0)
            Console.Write("Extended Midy's " +
                             "theorem holds!");       
        else
            Console.Write("Extended Midy's " +
                             "theorem doesn't hold!");       
    }
    else if (l % 2 != 0)
    {
        Console.Write("The repeating decimal is of " +
                         "odd length thus Extended Midy's " +
                         "theorem is not applicable");
    }
    else if (l % m != 0)
    {
        Console.Write("The repeating decimal can " +
                         "not be divided into m digits");
    }
}
 
// Driver code
public static void Main(string []args)
{
    int numr = 1, denr = 17, m = 4;
    string res = fractionToDecimal(numr, denr);
     
    if (res == "-1")
        Console.Write("The fraction does not " +
                         "have repeating decimal");
    else
    {
        Console.WriteLine("Repeating decimal = " + res);
        ExtendedMidys(res, denr, m);
    }
}
}
 
// This code is contributed by pratham76.

Javascript




// JavaScript program to demonstrate extended
// Midy's theorem
 
// Returns repeating sequence of a fraction.
// If repeating sequence doesn't exits,
// then returns -1
function fractionToDecimal(numerator, denominator)
{
    let res = "";
 
    /* Create a map to store already seen remainders
    remainder is used as key and its position in
    result is stored as value. Note that we need
    position for cases like 1/6. In this case,
    the recurring sequence doesn't start from first
    remainder. */
    let mp = {};
 
    // Find first remainder
    let rem = numerator % denominator;
 
    // Keep finding remainder until either remainder
    // becomes 0 or repeats
    while ((rem != 0) && (!mp.hasOwnProperty(rem))) {
 
        // Store this remainder
        mp[rem] = res.length;
 
        // Multiply remainder with 10
        rem = rem * 10;
 
        // Append rem / denr to result
        let res_part = Math.floor(rem / denominator);
        res += res_part.toString();
 
        // Update remainder
        rem = rem % denominator;
    }
 
    return (rem == 0) ? "-1" : res.substr(mp[rem]);
}
 
// Checks whether a number is prime or not
function isPrime(n)
{
    for (var i = 2; i <= n / 2; i++)
        if (n % i == 0)
            return false;
    return true;
}
 
// If all conditions are met,
// it proves Extended Midy's theorem
function ExtendedMidys(str, n, m)
{
    if (!isPrime(n)) {
        console.log(
            "Denominator is not prime, thus Extended Midy's theorem is not applicable");
        return;
    }
 
    let l = str.length;
    let part1 = 0, part2 = 0;
    if (l % 2 == 0 && l % m == 0) {
 
        // Dividing repeated part into m parts
        let part = new Array(m).fill(0);
        let sum = 0, res = 0;
        for (var i = 0; i < l; i++) {
            var var_ = Math.floor(i / m);
            part[var_] = part[var_] * 10 + parseInt(str[i]);
        }
 
        // Computing sum of parts.
        for (var i = 0; i < m; i++) {
            sum = sum + part[i];
            process.stdout.write(part[i] + " ");
        }
 
        // Checking for Extended Midy
        console.log();
        res = Math.pow(10, m) - 1;
        if (sum % res == 0)
            console.log("Extended Midy's theorem holds!");
        else
            console.log(
                "Extended Midy's theorem doesn't hold!");
    }
    else if (l % 2 != 0) {
        console.log(
            "The repeating decimal is of odd length thus Extended Midy's theorem is not applicable");
    }
    else if (l % m != 0) {
        console.log(
            "The repeating decimal can not be divided into m digits");
    }
}
 
// Driver code
let numr = 1, denr = 17, m = 4;
let res = fractionToDecimal(numr, denr);
if (res == "-1")
    console.log(
        "The fraction does not have repeating decimal");
else {
    console.log("Repeating decimal = " + res);
    ExtendedMidys(res, denr, m);
}
 
// This code is contributed by phasing17

Output: 

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

 


My Personal Notes arrow_drop_up
Recommended Articles
Page :

Start Your Coding Journey Now!