# Print all prime factors and their powers

Given a number N, print all its unique prime factors and their powers in N. **Examples:**

Input: N = 100 Output: Factor Power 2 2 5 2 Input: N = 35 Output: Factor Power 5 1 7 1

A **Simple Solution **is to first find prime factors of N. Then for every prime factor, find the highest power of it that divides N and print it.

An **Efficient Solution** is to use Sieve of Eratosthenes.

1)First compute an array s[N+1] using Sieve of Eratosthenes. s[i] = Smallest prime factor of "i" that divides "i". For example let N = 10 s[2] = s[4] = s[6] = s[8] = s[10] = 2; s[3] = s[9] = 3; s[5] = 5; s[7] = 7;2)Using the above computed array s[], we can find all powers in O(Log N) time. curr = s[N]; // Current prime factor of N cnt = 1; // Power of current prime factor // Printing prime factors and their powerswhile(N > 1) { N/=s[N]; // N is now N/s[N]. If new N also has its // smallest prime factor as curr, increment // power and continueif(curr == s[N]) { cnt++;continue;} // Print prime factor and its power

Below is the implementation of above steps.

## C++

`// C++ Program to print prime factors and their` `// powers using Sieve Of Eratosthenes` `#include<bits/stdc++.h>` `using` `namespace` `std;` `// Using SieveOfEratosthenes to find smallest prime` `// factor of all the numbers.` `// For example, if N is 10,` `// s[2] = s[4] = s[6] = s[10] = 2` `// s[3] = s[9] = 3` `// s[5] = 5` `// s[7] = 7` `void` `sieveOfEratosthenes(` `int` `N, ` `int` `s[])` `{` ` ` `// Create a boolean array "prime[0..n]" and` ` ` `// initialize all entries in it as false.` ` ` `vector <` `bool` `> prime(N+1, ` `false` `);` ` ` `// Initializing smallest factor equal to 2` ` ` `// for all the even numbers` ` ` `for` `(` `int` `i=2; i<=N; i+=2)` ` ` `s[i] = 2;` ` ` `// For odd numbers less then equal to n` ` ` `for` `(` `int` `i=3; i<=N; i+=2)` ` ` `{` ` ` `if` `(prime[i] == ` `false` `)` ` ` `{` ` ` `// s(i) for a prime is the number itself` ` ` `s[i] = i;` ` ` `// For all multiples of current prime number` ` ` `for` `(` `int` `j=i; j*i<=N; j+=2)` ` ` `{` ` ` `if` `(prime[i*j] == ` `false` `)` ` ` `{` ` ` `prime[i*j] = ` `true` `;` ` ` `// i is the smallest prime factor for` ` ` `// number "i*j".` ` ` `s[i*j] = i;` ` ` `}` ` ` `}` ` ` `}` ` ` `}` `}` `// Function to generate prime factors and its power` `void` `generatePrimeFactors(` `int` `N)` `{` ` ` `// s[i] is going to store smallest prime factor` ` ` `// of i.` ` ` `int` `s[N+1];` ` ` `// Filling values in s[] using sieve` ` ` `sieveOfEratosthenes(N, s);` ` ` `printf` `(` `"Factor Power\n"` `);` ` ` `int` `curr = s[N]; ` `// Current prime factor of N` ` ` `int` `cnt = 1; ` `// Power of current prime factor` ` ` `// Printing prime factors and their powers` ` ` `while` `(N > 1)` ` ` `{` ` ` `N /= s[N];` ` ` `// N is now N/s[N]. If new N als has smallest` ` ` `// prime factor as curr, increment power` ` ` `if` `(curr == s[N])` ` ` `{` ` ` `cnt++;` ` ` `continue` `;` ` ` `}` ` ` `printf` `(` `"%d\t%d\n"` `, curr, cnt);` ` ` `// Update current prime factor as s[N] and` ` ` `// initializing count as 1.` ` ` `curr = s[N];` ` ` `cnt = 1;` ` ` `}` `}` `//Driver Program` `int` `main()` `{` ` ` `int` `N = 360;` ` ` `generatePrimeFactors(N);` ` ` `return` `0;` `}` |

## Java

`// Java Program to print prime` `// factors and their powers using` `// Sieve Of Eratosthenes` `class` `GFG` `{` `// Using SieveOfEratosthenes` `// to find smallest prime` `// factor of all the numbers.` `// For example, if N is 10,` `// s[2] = s[4] = s[6] = s[10] = 2` `// s[3] = s[9] = 3` `// s[5] = 5` `// s[7] = 7` `static` `void` `sieveOfEratosthenes(` `int` `N,` ` ` `int` `s[])` `{` ` ` `// Create a boolean array` ` ` `// "prime[0..n]" and initialize` ` ` `// all entries in it as false.` ` ` `boolean` `[] prime = ` `new` `boolean` `[N + ` `1` `];` ` ` `// Initializing smallest` ` ` `// factor equal to 2` ` ` `// for all the even numbers` ` ` `for` `(` `int` `i = ` `2` `; i <= N; i += ` `2` `)` ` ` `s[i] = ` `2` `;` ` ` `// For odd numbers less` ` ` `// then equal to n` ` ` `for` `(` `int` `i = ` `3` `; i <= N; i += ` `2` `)` ` ` `{` ` ` `if` `(prime[i] == ` `false` `)` ` ` `{` ` ` `// s(i) for a prime is` ` ` `// the number itself` ` ` `s[i] = i;` ` ` `// For all multiples of` ` ` `// current prime number` ` ` `for` `(` `int` `j = i; j * i <= N; j += ` `2` `)` ` ` `{` ` ` `if` `(prime[i * j] == ` `false` `)` ` ` `{` ` ` `prime[i * j] = ` `true` `;` ` ` `// i is the smallest prime` ` ` `// factor for number "i*j".` ` ` `s[i * j] = i;` ` ` `}` ` ` `}` ` ` `}` ` ` `}` `}` `// Function to generate prime` `// factors and its power` `static` `void` `generatePrimeFactors(` `int` `N)` `{` ` ` `// s[i] is going to store` ` ` `// smallest prime factor of i.` ` ` `int` `[] s = ` `new` `int` `[N + ` `1` `];` ` ` `// Filling values in s[] using sieve` ` ` `sieveOfEratosthenes(N, s);` ` ` `System.out.println(` `"Factor Power"` `);` ` ` `int` `curr = s[N]; ` `// Current prime factor of N` ` ` `int` `cnt = ` `1` `; ` `// Power of current prime factor` ` ` `// Printing prime factors` ` ` `// and their powers` ` ` `while` `(N > ` `1` `)` ` ` `{` ` ` `N /= s[N];` ` ` `// N is now N/s[N]. If new N` ` ` `// also has smallest prime` ` ` `// factor as curr, increment power` ` ` `if` `(curr == s[N])` ` ` `{` ` ` `cnt++;` ` ` `continue` `;` ` ` `}` ` ` `System.out.println(curr + ` `"\t"` `+ cnt);` ` ` `// Update current prime factor` ` ` `// as s[N] and initializing` ` ` `// count as 1.` ` ` `curr = s[N];` ` ` `cnt = ` `1` `;` ` ` `}` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `N = ` `360` `;` ` ` `generatePrimeFactors(N);` `}` `}` `// This code is contributed by mits` |

## Python3

`# Python3 program to print prime` `# factors and their powers` `# using Sieve Of Eratosthenes` `# Using SieveOfEratosthenes to` `# find smallest prime factor` `# of all the numbers.` `# For example, if N is 10,` `# s[2] = s[4] = s[6] = s[10] = 2` `# s[3] = s[9] = 3` `# s[5] = 5` `# s[7] = 7` `def` `sieveOfEratosthenes(N, s):` ` ` ` ` `# Create a boolean array` ` ` `# "prime[0..n]" and initialize` ` ` `# all entries in it as false.` ` ` `prime ` `=` `[` `False` `] ` `*` `(N` `+` `1` `)` ` ` `# Initializing smallest factor` ` ` `# equal to 2 for all the even` ` ` `# numbers` ` ` `for` `i ` `in` `range` `(` `2` `, N` `+` `1` `, ` `2` `):` ` ` `s[i] ` `=` `2` ` ` `# For odd numbers less then` ` ` `# equal to n` ` ` `for` `i ` `in` `range` `(` `3` `, N` `+` `1` `, ` `2` `):` ` ` `if` `(prime[i] ` `=` `=` `False` `):` ` ` ` ` `# s(i) for a prime is` ` ` `# the number itself` ` ` `s[i] ` `=` `i` ` ` `# For all multiples of` ` ` `# current prime number` ` ` `for` `j ` `in` `range` `(i, ` `int` `(N ` `/` `i) ` `+` `1` `, ` `2` `):` ` ` `if` `(prime[i` `*` `j] ` `=` `=` `False` `):` ` ` `prime[i` `*` `j] ` `=` `True` ` ` `# i is the smallest` ` ` `# prime factor for` ` ` `# number "i*j".` ` ` `s[i ` `*` `j] ` `=` `i` `# Function to generate prime` `# factors and its power` `def` `generatePrimeFactors(N):` ` ` `# s[i] is going to store` ` ` `# smallest prime factor` ` ` `# of i.` ` ` `s ` `=` `[` `0` `] ` `*` `(N` `+` `1` `)` ` ` `# Filling values in s[]` ` ` `# using sieve` ` ` `sieveOfEratosthenes(N, s)` ` ` `print` `(` `"Factor Power"` `)` ` ` `# Current prime factor of N` ` ` `curr ` `=` `s[N]` ` ` ` ` `# Power of current prime factor` ` ` `cnt ` `=` `1` ` ` `# Printing prime factors and` ` ` `#their powers` ` ` `while` `(N > ` `1` `):` ` ` `N ` `/` `/` `=` `s[N]` ` ` `# N is now N/s[N]. If new N` ` ` `# also has smallest prime` ` ` `# factor as curr, increment` ` ` `# power` ` ` `if` `(curr ` `=` `=` `s[N]):` ` ` `cnt ` `+` `=` `1` ` ` `continue` ` ` `print` `(` `str` `(curr) ` `+` `"\t"` `+` `str` `(cnt))` ` ` `# Update current prime factor` ` ` `# as s[N] and initializing` ` ` `# count as 1.` ` ` `curr ` `=` `s[N]` ` ` `cnt ` `=` `1` `#Driver Program` `N ` `=` `360` `generatePrimeFactors(N)` `# This code is contributed by Ansu Kumari` |

## C#

`// C# Program to print prime` `// factors and their powers using` `// Sieve Of Eratosthenes` `class` `GFG` `{` `// Using SieveOfEratosthenes` `// to find smallest prime` `// factor of all the numbers.` `// For example, if N is 10,` `// s[2] = s[4] = s[6] = s[10] = 2` `// s[3] = s[9] = 3` `// s[5] = 5` `// s[7] = 7` `static` `void` `sieveOfEratosthenes(` `int` `N, ` `int` `[] s)` `{` ` ` `// Create a boolean array` ` ` `// "prime[0..n]" and initialize` ` ` `// all entries in it as false.` ` ` `bool` `[] prime = ` `new` `bool` `[N + 1];` ` ` `// Initializing smallest` ` ` `// factor equal to 2` ` ` `// for all the even numbers` ` ` `for` `(` `int` `i = 2; i <= N; i += 2)` ` ` `s[i] = 2;` ` ` `// For odd numbers less` ` ` `// then equal to n` ` ` `for` `(` `int` `i = 3; i <= N; i += 2)` ` ` `{` ` ` `if` `(prime[i] == ` `false` `)` ` ` `{` ` ` `// s(i) for a prime is` ` ` `// the number itself` ` ` `s[i] = i;` ` ` `// For all multiples of` ` ` `// current prime number` ` ` `for` `(` `int` `j = i; j * i <= N; j += 2)` ` ` `{` ` ` `if` `(prime[i * j] == ` `false` `)` ` ` `{` ` ` `prime[i * j] = ` `true` `;` ` ` `// i is the smallest prime` ` ` `// factor for number "i*j".` ` ` `s[i * j] = i;` ` ` `}` ` ` `}` ` ` `}` ` ` `}` `}` `// Function to generate prime` `// factors and its power` `static` `void` `generatePrimeFactors(` `int` `N)` `{` ` ` `// s[i] is going to store` ` ` `// smallest prime factor of i.` ` ` `int` `[] s = ` `new` `int` `[N + 1];` ` ` `// Filling values in s[] using sieve` ` ` `sieveOfEratosthenes(N, s);` ` ` `System.Console.WriteLine(` `"Factor Power"` `);` ` ` `int` `curr = s[N]; ` `// Current prime factor of N` ` ` `int` `cnt = 1; ` `// Power of current prime factor` ` ` `// Printing prime factors` ` ` `// and their powers` ` ` `while` `(N > 1)` ` ` `{` ` ` `N /= s[N];` ` ` `// N is now N/s[N]. If new N` ` ` `// also has smallest prime` ` ` `// factor as curr, increment power` ` ` `if` `(curr == s[N])` ` ` `{` ` ` `cnt++;` ` ` `continue` `;` ` ` `}` ` ` `System.Console.WriteLine(curr + ` `"\t"` `+ cnt);` ` ` `// Update current prime factor` ` ` `// as s[N] and initializing` ` ` `// count as 1.` ` ` `curr = s[N];` ` ` `cnt = 1;` ` ` `}` `}` `// Driver Code` `static` `void` `Main()` `{` ` ` `int` `N = 360;` ` ` `generatePrimeFactors(N);` `}` `}` `// This code is contributed by mits` |

## PHP

`<?php` `// PHP Program to print prime factors and` `// their powers using Sieve Of Eratosthenes` `// Using SieveOfEratosthenes to find smallest` `// prime factor of all the numbers.` `// For example, if N is 10,` `// s[2] = s[4] = s[6] = s[10] = 2` `// s[3] = s[9] = 3` `// s[5] = 5` `// s[7] = 7` `function` `sieveOfEratosthenes(` `$N` `, &` `$s` `)` `{` ` ` `// Create a boolean array "prime[0..n]" and` ` ` `// initialize all entries in it as false.` ` ` `$prime` `= ` `array_fill` `(0, ` `$N` `+ 1, false);` ` ` `// Initializing smallest factor equal` ` ` `// to 2 for all the even numbers` ` ` `for` `(` `$i` `= 2; ` `$i` `<= ` `$N` `; ` `$i` `+= 2)` ` ` `$s` `[` `$i` `] = 2;` ` ` `// For odd numbers less then equal to n` ` ` `for` `(` `$i` `= 3; ` `$i` `<= ` `$N` `; ` `$i` `+= 2)` ` ` `{` ` ` `if` `(` `$prime` `[` `$i` `] == false)` ` ` `{` ` ` `// s(i) for a prime is the` ` ` `// number itself` ` ` `$s` `[` `$i` `] = ` `$i` `;` ` ` `// For all multiples of current` ` ` `// prime number` ` ` `for` `(` `$j` `= ` `$i` `; ` `$j` `* ` `$i` `<= ` `$N` `; ` `$j` `+= 2)` ` ` `{` ` ` `if` `(` `$prime` `[` `$i` `* ` `$j` `] == false)` ` ` `{` ` ` `$prime` `[` `$i` `* ` `$j` `] = true;` ` ` `// i is the smallest prime factor` ` ` `// for number "i*j".` ` ` `$s` `[` `$i` `* ` `$j` `] = ` `$i` `;` ` ` `}` ` ` `}` ` ` `}` ` ` `}` `}` `// Function to generate prime factors` `// and its power` `function` `generatePrimeFactors(` `$N` `)` `{` ` ` `// s[i] is going to store smallest` ` ` `// prime factor of i.` ` ` `$s` `= ` `array_fill` `(0, ` `$N` `+ 1, 0);` ` ` `// Filling values in s[] using sieve` ` ` `sieveOfEratosthenes(` `$N` `, ` `$s` `);` ` ` `print` `(` `"Factor Power\n"` `);` ` ` `$curr` `= ` `$s` `[` `$N` `]; ` `// Current prime factor of N` ` ` `$cnt` `= 1; ` `// Power of current prime factor` ` ` `// Printing prime factors and their powers` ` ` `while` `(` `$N` `> 1)` ` ` `{` ` ` `if` `(` `$s` `[` `$N` `])` ` ` `$N` `= (int)(` `$N` `/ ` `$s` `[` `$N` `]);` ` ` `// N is now N/s[N]. If new N als has smallest` ` ` `// prime factor as curr, increment power` ` ` `if` `(` `$curr` `== ` `$s` `[` `$N` `])` ` ` `{` ` ` `$cnt` `++;` ` ` `continue` `;` ` ` `}` ` ` `print` `(` `$curr` `. ` `"\t"` `. ` `$cnt` `. ` `"\n"` `);` ` ` `// Update current prime factor as s[N]` ` ` `// and initializing count as 1.` ` ` `$curr` `= ` `$s` `[` `$N` `];` ` ` `$cnt` `= 1;` ` ` `}` `}` `// Driver Code` `$N` `= 360;` `generatePrimeFactors(` `$N` `);` `// This code is contributed by mits` `?>` |

## Javascript

`<script>` `// javascript Program to print prime` `// factors and their powers using` `// Sieve Of Eratosthenes` `// Using SieveOfEratosthenes` `// to find smallest prime` `// factor of all the numbers.` `// For example, if N is 10,` `// s[2] = s[4] = s[6] = s[10] = 2` `// s[3] = s[9] = 3` `// s[5] = 5` `// s[7] = 7` `function` `sieveOfEratosthenes(N, s)` `{` ` ` `// Create a boolean array` ` ` `// "prime[0..n]" and initialize` ` ` `// all entries in it as false.` ` ` `prime = Array.from({length: N+1}, (_, i) => ` `false` `);` ` ` `// Initializing smallest` ` ` `// factor equal to 2` ` ` `// for all the even numbers` ` ` `for` `(i = 2; i <= N; i += 2)` ` ` `s[i] = 2;` ` ` `// For odd numbers less` ` ` `// then equal to n` ` ` `for` `(i = 3; i <= N; i += 2)` ` ` `{` ` ` `if` `(prime[i] == ` `false` `)` ` ` `{` ` ` `// s(i) for a prime is` ` ` `// the number itself` ` ` `s[i] = i;` ` ` `// For all multiples of` ` ` `// current prime number` ` ` `for` `(j = i; j * i <= N; j += 2)` ` ` `{` ` ` `if` `(prime[i * j] == ` `false` `)` ` ` `{` ` ` `prime[i * j] = ` `true` `;` ` ` `// i is the smallest prime` ` ` `// factor for number "i*j".` ` ` `s[i * j] = i;` ` ` `}` ` ` `}` ` ` `}` ` ` `}` `}` `// Function to generate prime` `// factors and its power` `function` `generatePrimeFactors(N)` `{` ` ` `// s[i] is going to store` ` ` `// smallest prime factor of i.` ` ` `var` `s = Array.from({length: N+1}, (_, i) => 0);` ` ` `// Filling values in s using sieve` ` ` `sieveOfEratosthenes(N, s);` ` ` `document.write(` `"Factor Power"` `);` ` ` `var` `curr = s[N]; ` `// Current prime factor of N` ` ` `var` `cnt = 1; ` `// Power of current prime factor` ` ` `// Printing prime factors` ` ` `// and their powers` ` ` `while` `(N > 1)` ` ` `{` ` ` `N /= s[N];` ` ` `// N is now N/s[N]. If new N` ` ` `// also has smallest prime` ` ` `// factor as curr, increment power` ` ` `if` `(curr == s[N])` ` ` `{` ` ` `cnt++;` ` ` `continue` `;` ` ` `}` ` ` `document.write(` `"<br>"` `+curr + ` `"\t"` `+ cnt);` ` ` `// Update current prime factor` ` ` `// as s[N] and initializing` ` ` `// count as 1.` ` ` `curr = s[N];` ` ` `cnt = 1;` ` ` `}` `}` `// Driver Code` `var` `N = 360;` `generatePrimeFactors(N);` `// This code contributed by Princi Singh` `</script>` |

**Output:**

Factor Power 2 3 3 2 5 1

The above algorithm finds all powers in O(Log N) time after we have filled s[]. This can be very useful in competitive environment where we have an upper limit and we need to compute prime factors and their powers for many test cases. In this scenario, the array needs to be s[] filled only once.

This article is contributed by **Rahul Agrawal**. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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