Given a fraction, find a recurring sequence of digits if it exists, otherwise, print “No recurring sequence”.
Examples:
Input : Numerator = 8, Denominator = 3 Output : Recurring sequence is 6 Explanation : 8/3 = 2.66666666....... Input : Numerator = 50, Denominator = 22 Output : Recurring sequence is 27 Explanation : 50/22 = 2.272727272..... Input : Numerator = 11, Denominator = 2 Output : No recurring sequence Explanation : 11/2 = 5.5
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When does the fractional part repeat?
Let us simulate the process of converting fractions to decimals. Let us look at the part where we have already figured out the integer part, which is floor(numerator/denominator). Now we are left with ( remainder = numerator%denominator ) / denominator.
If you remember the process of converting to decimal, at each step we do the following :
- Multiply the remainder by 10.
- Append the remainder/denominator to the result.
- Remainder = remainder % denominator.
At any moment, if the remainder becomes 0, we are done.
However, when there is a recurring sequence, the remainder never becomes 0. For example, if you look at 1/3, the remainder never becomes 0.
Below is one important observation :
If we start with the remainder ‘rem’ and if the remainder repeats at any point in time, the digits between the two occurrences of ‘rem’ keep repeating.
So the idea is to store seen remainders in a map. Whenever a remainder repeats, we return the sequence before the next occurrence.
Below is the implementation of the above idea.
// C++ program to find repeating // sequence in a fraction #include <bits/stdc++.h> using namespace std;
// This function returns repeating sequence of // a fraction. If repeating sequence doesn't // exist, then returns empty string string fractionToDecimal( int numr, int denr)
{ string res; // Initialize result
// 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.
map< int , int > mp;
mp.clear();
// Find first remainder
int rem = numr % denr;
// 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 / denr;
res += to_string(res_part);
// Update remainder
rem = rem % denr;
}
return (rem == 0) ? "" : res.substr(mp[rem]);
} // Driver code int main()
{ int numr = 50, denr = 22;
string res = fractionToDecimal(numr, denr);
if (res == "" )
cout << "No recurring sequence" ;
else
cout << "Recurring sequence is " << res;
return 0;
} |
// Java program to find // repeating sequence // in a fraction import java.util.*;
class GFG {
// This function returns repeating
// sequence of a fraction. If
// repeating sequence doesn't
// exist, then returns empty String
static String fractionToDecimal( int numr, int denr)
{
// Initialize result
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<>();
mp.clear();
// Find first remainder
int rem = numr % denr;
// 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 / denr;
res += String.valueOf(res_part);
// Update remainder
rem = rem % denr;
}
if (rem == 0 )
return "" ;
else if (mp.containsKey(rem))
return res.substring(mp.get(rem));
return "" ;
}
// Driver code
public static void main(String[] args)
{
int numr = 50 , denr = 22 ;
String res = fractionToDecimal(numr, denr);
if (res == "" )
System.out.print( "No recurring sequence" );
else
System.out.print( "Recurring sequence is "
+ res);
}
} // This code is contributed by gauravrajput1 |
# Python3 program to find repeating # sequence in a fraction # This function returns repeating sequence # of a fraction.If repeating sequence doesn't # exist, then returns empty string def fractionToDecimal(numr, denr):
# Initialize result
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 = {}
# Find first remainder
rem = numr % denr
# 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 / / denr
res + = str (res_part)
# Update remainder
rem = rem % denr
if (rem = = 0 ):
return ""
else :
return res[mp[rem]:]
# Driver code numr, denr = 50 , 22
res = fractionToDecimal(numr, denr)
if (res = = ""):
print ( "No recurring sequence" )
else :
print ( "Recurring sequence is" , res)
# This code is contributed by divyeshrabadiya07 |
// C# program to find repeating sequence // in a fraction using System;
using System.Collections.Generic;
class GFG {
// This function returns repeating
// sequence of a fraction. If
// repeating sequence doesn't
// exist, then returns empty String
static string fractionToDecimal( int numr, int denr)
{
// Initialize result
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 = numr % denr;
// 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 / denr;
res += res_part.ToString();
// Update remainder
rem = rem % denr;
}
if (rem == 0)
return "" ;
else if (mp.ContainsKey(rem))
return res.Substring(mp[rem]);
return "" ;
}
// Driver code
public static void Main( string [] args)
{
int numr = 50, denr = 22;
string res = fractionToDecimal(numr, denr);
if (res == "" )
Console.Write( "No recurring sequence" );
else
Console.Write( "Recurring sequence is " + res);
}
} // This code is contributed by rutvik_56 |
<script> // Javascript program to find // repeating sequence // in a fraction // This function returns repeating
// sequence of a fraction. If
// repeating sequence doesn't
// exist, then returns empty String
function fractionToDecimal(numr, denr)
{
// Initialize result
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 = new Map();
mp.clear();
// Find first remainder
let rem = numr % denr;
// Keep finding remainder until
// either remainder becomes 0 or repeats
while ((rem != 0) && (!mp.has(rem)))
{
// Store this remainder
mp.set(rem, res.length);
// Multiply remainder with 10
rem = rem * 10;
// Append rem / denr to result
let res_part = Math.floor(rem / denr);
res += res_part.toString();
// Update remainder
rem = rem % denr;
}
if (rem == 0)
return "" ;
else if (mp.has(rem))
return res.substr(mp.get(rem));
return "" ;
}
// Driver program let numr = 50, denr = 22;
let res = fractionToDecimal(numr, denr);
if (res == "" )
document.write( "No recurring sequence" );
else
document.write( "Recurring sequence is "
+ res);
</script> |
Recurring sequence is 27
Time Complexity : O(N)
Auxiliary Space : O(N) , as we use map as extra space.