search
HomeBackend DevelopmentC++Checks whether the given two numbers are a friendly pair

Checks whether the given two numbers are a friendly pair

Friendly numbers − According to number theory, a friendly number is two or more numbers with the same abundance index.

Richness Index - The richness index of a natural number can be defined as the ratio between the sum of all divisors of a natural number and the natural number itself.

The abundance of number n can be expressed as $\mathrm{\frac{\sigma(n)}{n}}$, where $\mathrm{\sigma(n)}$ represents the divisor function equal to the approximation of all n number.

For example, the abundance index of the natural number 30 is,

$$\mathrm{\frac{\sigma(30)}{30}=\frac{1 2 3 5 6 10 15 30}{30}=\frac{72}{ 30}=\frac{12 }{5}}$$

If there is a number m mn, then the number n is called a "friendly number".

$\mathrm{\frac{\sigma(m)}{m}=\frac{\sigma(n)}{n}}$

Friendly Pair − Two numbers with the same surplus index are called "friendly pairs".

Problem Statement

Given two numbers Num1 and Num2. Returns if the two numbers are not a friendly pair.

Example 1

Input: Num1 = 30, Num2 = 140
Output: Yes
The Chinese translation of

Explanation

is:

Explanation

$$\mathrm{\frac{\sigma(30)}{30}=\frac{1 2 3 5 6 10 15 30}{30}=\frac{72}{ 30}=\frac{12 }{5}}$$

$$\mathrm{\frac{\sigma(140)}{140}=\frac{1 2 4 5 7 10 14 20 28 35 70 140}{140 }=\frac{336}{140}= \frac{12}{5}}$$

Since,\frac{\sigma(30)}{30}=\frac{\sigma(140)}{140}, 30 and 140 are a pair of friendly numbers.

Example Example 2

Input: Num1 = 5, Num2 = 24
Output: No
The Chinese translation of

Explanation

is:

Explanation

$$\mathrm{\frac{\sigma(5)}{5}=\frac{1 5}{5}=\frac{6}{5}=\frac{6}{5}} $ $

$$\mathrm{\frac{\sigma(24)}{24}=\frac{1 2 3 4 6 8 12 24}{24}=\frac{60}{ 24}=\frac{15 }{6}}$$

Since $\mathrm{\frac{\sigma(5)}{5}\neq\frac{\sigma(24)}{24}}$, 5 and 24 are not a friendly pair. p>

Method 1: Brute force method

The brute force way to solve this problem is to first find the sum of all divisors of the two numbers, then calculate the value of their abundance index, and compare to get the result.

pseudocode

procedure sumOfDivisors (n)
   sum = 0
   for i = 1 to n
      if i is a factor of n
         sum = sum + i
      end if
   ans = sum
end procedure

procedure friendlyPair (num1, num2)
   sum1 = sumOfDivisors (num1)
   sum2 = sumOfDivisors (num2)
   abIndex1 = sum1 / num1
   abIndex2 = sum2 / num2
   if (abIndex1 == abIndex2)
      ans = TRUE
   else
      ans = FALSE
   end if
end procedure

Example: C implementation

In the following program, the sum of all divisors is calculated to find the abundance index.

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

// Function to find sum of all the divisors of number n
int sumOfDivisors(int n){
   int sum = 0;
   for (int i = 1; i <= n; i++){
      if (n % i == 0){
         sum += i;
      }
   }
   return sum;
}
// Function to find if two numbers are friendly pairs or not
int friendlyPair(int num1, int num2){

   // Finding the sum of all divisors of num1 and num2
   int sum1 = sumOfDivisors(num1);
   int sum2 = sumOfDivisors(num2);
   
   // Calculating the abundancy index as the ratio of the sum of divisors by the number
   int abIn1 = sum1 / num1, abIn2 = sum2 / num2;
   
   // Friendly pair if the abundancy index of both the numbers are same
   if (abIn1 == abIn2){
      return true;
   }
   return false;
}
int main(){
   int num1 = 30, num2 = 140;
   cout << num1 << " and " << num2 << " are friendly pair : ";
   if (friendlyPair(num1, num2)){
      cout << "YES";
   }
   else{
      cout << "NO";
   }
   return 0;
} 

Output

30 and 140 are friendly pair : YES

Time complexity - O(n), because the sumOfDivisors() function traverses a loop

Space complexity - O(1)

Method 2: Reduced form of abundance index

The simplified form of the richness index can be found by dividing both the numerator and denominator by the greatest common divisor. We then check whether the two numbers are a friendly pair by checking whether the reduced forms of richness are equal, i.e. checking whether their numerators and denominators are equal.

pseudocode

procedure sumOfDivisors (n)
   ans = 1
   for i = 1 to sqrt(n)
      count = 0
      sum = 1
      term = 1
      while n % i == 0
         count = count + 1
         n = n / i
         term = term * i
         sum = sum + term
      ans = ans * sum
   if n >= 2
      ans = ans * (n + 1)
   end if
end procedure

procedure gcd (n1, n2)
   if n1 == 0
      return n2
   end if
   rem = n2 % n1
   return gcd (rem, n2)
end procedure

procedure friendlyPair (num1, num2)
   sum1 = sumOfDivisors (num1)
   sum2 = sumOfDivisors (num2)
   gcd1 = gcd (num1, sum1)
   gcd2 = gcd (num2, sum2)
   if (num1 / gcd1 == num2 / gcd2) && (sum1 / gcd1 == sum2 / gcd2)
      ans = TRUE
   else
      ans = FALSE
   end if
end procedure

Example: C implementation

In the following program, we check whether the abundance index of the reduced form of two numbers is the same by comparing the numerator and denominator.

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

// Function to find the sum of all the divisors of number n
int sumOfDivisors(int n){
   int ans = 1;
   
   // By looping till sqrt(n), we traverse all the prime factors of n
   for (int i = 2; i <= sqrt(n); i++){
      int cnt = 0, sum = 1, term = 1;
      while (n % i == 0){
         cnt++;
         
         // Reducing the value of n
         n /= i;
         term *= i;
         sum += term;
      }
      ans *= sum;
   }
   
   // When n is a prime number greater than 2
   if (n >= 2){
      ans *= (n + 1);
   }
   return ans;
}

// Function to find the gcd of two numbers
int gcd(int num1, int num2){
   if (num1 == 0) {
      return num2;
   }
   int rem = num2 % num1;
   return gcd(rem, num1);
}

// Function to find if two numbers are friendly pairs or not
int friendlyPair(int num1, int num2){

   // Finding the sum of all divisors of num1 and num2
   int sum1 = sumOfDivisors(num1);
   int sum2 = sumOfDivisors(num2);
   
   // Finding gcd of num and the sum of its divisors
   int gcd1 = gcd(num1, sum1);
   int gcd2 = gcd(num2, sum2);
   
   // Checking if the numerator and denominator of the reduced abundancy index are the same or not
   if (((num1 / gcd1) == (num2 / gcd2)) && ((sum1 / gcd1) == (sum2 / gcd2))){
      return true;
   }
   return false;
}
int main(){
   int num1 = 30, num2 = 140;
   cout << num1 << " and " << num2 << " are friendly pair : ";
   if (friendlyPair(num1, num2)){
      cout << "YES";
   }
   else{
      cout << "NO";
   }
   return 0;
}

Output

30 and 140 are friendly pair : YES

Time complexity - The time complexity of the sumOfDivisors() function is O(n1/2log2n).

Space complexity - O(1)

in conclusion

To sum up, a friendly pair refers to two natural numbers with the same abundance index, that is, the ratio of the sum of all divisors of the number to the number itself. To find whether two numbers are a friendly pair, follow the approach above, specifying a brute-force solution with time complexity O(n) and an optimized solution with time complexity O(n1/2log2n) plan.

The above is the detailed content of Checks whether the given two numbers are a friendly pair. For more information, please follow other related articles on the PHP Chinese website!

Statement
This article is reproduced at:tutorialspoint. If there is any infringement, please contact admin@php.cn delete
The Longevity of C  : Examining Its Current StatusThe Longevity of C : Examining Its Current StatusApr 26, 2025 am 12:02 AM

C is still important in modern programming because of its efficient, flexible and powerful nature. 1)C supports object-oriented programming, suitable for system programming, game development and embedded systems. 2) Polymorphism is the highlight of C, allowing the call to derived class methods through base class pointers or references to enhance the flexibility and scalability of the code.

C# vs. C   Performance: Benchmarking and ConsiderationsC# vs. C Performance: Benchmarking and ConsiderationsApr 25, 2025 am 12:25 AM

The performance differences between C# and C are mainly reflected in execution speed and resource management: 1) C usually performs better in numerical calculations and string operations because it is closer to hardware and has no additional overhead such as garbage collection; 2) C# is more concise in multi-threaded programming, but its performance is slightly inferior to C; 3) Which language to choose should be determined based on project requirements and team technology stack.

C  : Is It Dying or Simply Evolving?C : Is It Dying or Simply Evolving?Apr 24, 2025 am 12:13 AM

C isnotdying;it'sevolving.1)C remainsrelevantduetoitsversatilityandefficiencyinperformance-criticalapplications.2)Thelanguageiscontinuouslyupdated,withC 20introducingfeatureslikemodulesandcoroutinestoimproveusabilityandperformance.3)Despitechallen

C   in the Modern World: Applications and IndustriesC in the Modern World: Applications and IndustriesApr 23, 2025 am 12:10 AM

C is widely used and important in the modern world. 1) In game development, C is widely used for its high performance and polymorphism, such as UnrealEngine and Unity. 2) In financial trading systems, C's low latency and high throughput make it the first choice, suitable for high-frequency trading and real-time data analysis.

C   XML Libraries: Comparing and Contrasting OptionsC XML Libraries: Comparing and Contrasting OptionsApr 22, 2025 am 12:05 AM

There are four commonly used XML libraries in C: TinyXML-2, PugiXML, Xerces-C, and RapidXML. 1.TinyXML-2 is suitable for environments with limited resources, lightweight but limited functions. 2. PugiXML is fast and supports XPath query, suitable for complex XML structures. 3.Xerces-C is powerful, supports DOM and SAX resolution, and is suitable for complex processing. 4. RapidXML focuses on performance and parses extremely fast, but does not support XPath queries.

C   and XML: Exploring the Relationship and SupportC and XML: Exploring the Relationship and SupportApr 21, 2025 am 12:02 AM

C interacts with XML through third-party libraries (such as TinyXML, Pugixml, Xerces-C). 1) Use the library to parse XML files and convert them into C-processable data structures. 2) When generating XML, convert the C data structure to XML format. 3) In practical applications, XML is often used for configuration files and data exchange to improve development efficiency.

C# vs. C  : Understanding the Key Differences and SimilaritiesC# vs. C : Understanding the Key Differences and SimilaritiesApr 20, 2025 am 12:03 AM

The main differences between C# and C are syntax, performance and application scenarios. 1) The C# syntax is more concise, supports garbage collection, and is suitable for .NET framework development. 2) C has higher performance and requires manual memory management, which is often used in system programming and game development.

C# vs. C  : History, Evolution, and Future ProspectsC# vs. C : History, Evolution, and Future ProspectsApr 19, 2025 am 12:07 AM

The history and evolution of C# and C are unique, and the future prospects are also different. 1.C was invented by BjarneStroustrup in 1983 to introduce object-oriented programming into the C language. Its evolution process includes multiple standardizations, such as C 11 introducing auto keywords and lambda expressions, C 20 introducing concepts and coroutines, and will focus on performance and system-level programming in the future. 2.C# was released by Microsoft in 2000. Combining the advantages of C and Java, its evolution focuses on simplicity and productivity. For example, C#2.0 introduced generics and C#5.0 introduced asynchronous programming, which will focus on developers' productivity and cloud computing in the future.

See all articles

Hot AI Tools

Undresser.AI Undress

Undresser.AI Undress

AI-powered app for creating realistic nude photos

AI Clothes Remover

AI Clothes Remover

Online AI tool for removing clothes from photos.

Undress AI Tool

Undress AI Tool

Undress images for free

Clothoff.io

Clothoff.io

AI clothes remover

Video Face Swap

Video Face Swap

Swap faces in any video effortlessly with our completely free AI face swap tool!

Hot Tools

MinGW - Minimalist GNU for Windows

MinGW - Minimalist GNU for Windows

This project is in the process of being migrated to osdn.net/projects/mingw, you can continue to follow us there. MinGW: A native Windows port of the GNU Compiler Collection (GCC), freely distributable import libraries and header files for building native Windows applications; includes extensions to the MSVC runtime to support C99 functionality. All MinGW software can run on 64-bit Windows platforms.

SAP NetWeaver Server Adapter for Eclipse

SAP NetWeaver Server Adapter for Eclipse

Integrate Eclipse with SAP NetWeaver application server.

Safe Exam Browser

Safe Exam Browser

Safe Exam Browser is a secure browser environment for taking online exams securely. This software turns any computer into a secure workstation. It controls access to any utility and prevents students from using unauthorized resources.

mPDF

mPDF

mPDF is a PHP library that can generate PDF files from UTF-8 encoded HTML. The original author, Ian Back, wrote mPDF to output PDF files "on the fly" from his website and handle different languages. It is slower than original scripts like HTML2FPDF and produces larger files when using Unicode fonts, but supports CSS styles etc. and has a lot of enhancements. Supports almost all languages, including RTL (Arabic and Hebrew) and CJK (Chinese, Japanese and Korean). Supports nested block-level elements (such as P, DIV),

Dreamweaver CS6

Dreamweaver CS6

Visual web development tools