Exercise #26: Prime Factorization by Zoran Horvat @zoranh75 March 09, 2014
Problem Statement
Given a positive integer number, write a function which returns an array with
all prime factors of that number.
Examples: If number 1064 is passed, the function should return array with
numbers 2, 2, 2, 7 and 19. If number 1 is passed to the function, an array
containing only value 1 should be returned.
Problem Analysis
Breaking the input number N into its prime constituents can be accomplished by attempting to divide it by
different candidate values. One of the difficulties that we are facing is then
to find proper candidates. First of all, they must be greater than one. If
input number is equal to one, then number one alone is the only factor and the
process ends. Otherwise, valid candidates will start from value 2. The upper
bound for candidates would then be the greatest number not exceeding square
root of N. It would be pointless to try to divide N by any number greater than its square root, because such factor would already
have been found earlier. For detailed analysis, please refer to exercise Finding All Prime Numbers Smaller Than Specified Value.
By this point we have established the domain from which possible prime factors
are drawn. It is the set of numbers between 2 and square root of N, inclusive. The factorization process would then go like this. As we take
candidates in turns, we try whether each of them divides N with no remainder. In case of success, we know that the real factor is found
and it is immediately added to the resulting array.
But then, two important consequences occur. First, number N should be divided by the current factor, so that the prime factor is removed
from it. (On a related note, we know that the factor is prime because all
smaller values have already been tried out.) This also means that upper bound
for candidates, the square root of N, reduces as well and the set of remaining candidates shrinks. Second
consequence is that we do not move to the next candidate. Instead, the same
candidate is retained, because it might divide N more than once.
Finally, after all candidates have been exhausted, number N will still be greater than one. That value would be the final prime factor,
the one that could not be split into smaller factors using division.
So this is the algorithm. Before putting it in motion, we can make one
additional observation. Apart from numbers 2 and 3, no other value divisible by
2 or 3 should be taken as candidate. Such numbers are obviously not prime and
cannot be promoted to the output. This means that only numbers in form 2k1 and 2k+1 remain, for any positive k, in addition to primes 2 and 3. This simple action reduces the candidates set
roughly by two thirds, improving the overall algorithm performance. (The same
idea was also used in exercise Finding All Prime Numbers Smaller Than Specified Value).
Now that everything is in place, we can write the pseudocode:
function GetPrimeFactors(n)
 n  positive number
begin
factors  empty array
if n = 1 then
factors = [1]
else
begin
factor = 2
step = 2
while factor * factor <= n
begin
if n mod factor = 0 then
append factor to factors
n = n / factor
else if factor < 3 then
factor = factor + 1
else if factor < 5 then
factor = factor + 2
else
factor = factor + step
step = 6  step
end
append n to factors
end
return factors
end
Implementation
Below is the console application in C# which lets the user enter number N and
then splits the number into its prime factors.
using System;
using System.Collections.Generic;
namespace PrimeFactorization
{
class Program
{
private static int[] GetPrimeFactors(int n)
{
List<int> factors = new List<int>();
if (n == 1)
factors.Add(1);
else
{
int factor = 2;
int step = 2;
while (factor * factor <= n)
{
if (n % factor == 0)
{
factors.Add(factor);
n = factor;
}
else if (factor < 3)
{
factor++;
}
else if (factor < 5)
{
factor += 2;
}
else
{
factor += step;
step = 6  step;
}
}
factors.Add(n);
}
return factors.ToArray();
}
static void Main()
{
while (true)
{
Console.Write("Enter positive number (zero to exit): ");
int n = int.Parse(Console.ReadLine());
if (n <= 0)
break;
int[] factors = GetPrimeFactors(n);
PrintFactors(n, factors);
}
}
private static void PrintFactors(int n, int[] factors)
{
Console.Write("{0} = {1}", n, factors[0]);
for (int i = 1; i < factors.Length; i++)
Console.Write(" * {0}", factors[i]);
Console.WriteLine();
Console.WriteLine();
}
}
}
Demonstration
When application above is run, it produces the following output:
Enter positive number (zero to exit): 1064
1064 = 2 * 2 * 2 * 7 * 19
Enter positive number (zero to exit): 574
574 = 2 * 7 * 41
Enter positive number (zero to exit): 912
912 = 2 * 2 * 2 * 2 * 3 * 19
Enter positive number (zero to exit): 9954
9954 = 2 * 3 * 3 * 7 * 79
Enter positive number (zero to exit): 0
See also:
Published: Mar 9, 2014; Modified: Dec 14, 2014
ZORAN HORVAT Zoran is software architect dedicated to clean design and CTO in a growing software company. Since 2014 Zoran is an author at Pluralsight where he is preparing a series of courses on objectoriented and functional design, design patterns, writing unit and integration tests and applying methods to improve code design and longterm maintainability. Follow him on Twitter @zoranh75 to receive updates and links to new articles. Watch Zoran's video courses at pluralsight.com (requires registration): Making Your C# Code More ObjectOriented This course will help leverage your conceptual understanding to produce proper objectoriented code, where objects will completely replace procedural code for the sake of flexibility and maintainability. More... Advanced Defensive Programming Techniques This course will lead you step by step through the process of developing defensive design practices, which can substitute common defensive coding, for the better of software design and implementation. More... Tactical Design Patterns in .NET: Creating Objects This course sheds light on issues that arise when implementing creational design patterns and then provides practical solutions that will make our code easier to write and more stable when running. More... Tactical Design Patterns in .NET: Managing Responsibilities Applying a design pattern to a realworld problem is not as straightforward as literature implicitly tells us. It is a more engaged process. This course gives an insight to tactical decisions we need to make when applying design patterns that have to do with separating and implementing class responsibilities. More... Tactical Design Patterns in .NET: Control Flow Improve your skills in writing simpler and safer code by applying coding practices and design patterns that are affecting control flow. More... Writing Highly Maintainable Unit Tests This course will teach you how to develop maintainable and sustainable tests as your production code grows and develops. More... Improving Testability Through Design This course tackles the issues of designing a complex application so that it can be covered with high quality tests. More... Share this article
