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    <p>Ok. So you said you understand the basics, but you're not sure EXACTLY how it works. First of all, this is a great answer to the Project Euler question it stems from. I've done a lot of research into this problem and this is by far the simplest response.</p> <p>For the purpose of explanation, I'll let <code>n = 20</code>. To run the real Project Euler problem, let <code>n = 600851475143</code>.</p> <pre><code>n = 20 i = 2 while i * i &lt; n: while n%i == 0: n = n / i i = i + 1 print (n) </code></pre> <p>This explanation uses two <code>while</code> loops. The biggest thing to remember about <code>while</code> loops is that they run until they are no longer <code>true</code>. </p> <p>The outer loop states that while <code>i * i</code> isn't greater than <code>n</code> (because the largest prime factor will never be larger than the square root of <code>n</code>), add <code>1</code> to <code>i</code> after the inner loop runs. </p> <p>The inner loop states that while <code>i</code> divides evenly into <code>n</code>, replace <code>n</code> with <code>n</code> divided by <code>i</code>. This loop runs continuously until it is no longer true. For <code>n=20</code> and <code>i=2</code>, <code>n</code> is replaced by <code>10</code>, then again by <code>5</code>. Because <code>2</code> doesn't evenly divide into <code>5</code>, the loop stops with <code>n=5</code> and the outer loop finishes, producing <code>i+1=3</code>.</p> <p>Finally, because <code>3</code> squared is greater than <code>5</code>, the outer loop is no longer <code>true</code> and prints the result of <code>n</code>.</p> <p>Thanks for posting this. I looked at the code forever before realizing how exactly it worked. Hopefully, this is what you're looking for in a response. If not, let me know and I can explain further.</p>
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    1. POPython Finding Prime Factors
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    1. USWill Luce
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    1. CO'because the largest prime factor will never be larger than the square root of n' - why? largest prime factor of 10 is 5, and 5 is greater than the square root of 10
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    2. COWhat about the case when `n=4`? This seems like it would print 4 as a prime
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    3. CO@Mathai I'm guessing Will meant the smallest prime factor, see: http://math.stackexchange.com/questions/102755/greatest-prime-factor-of-n-is-less-than-square-root-of-n-proof
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