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    <p>Provided your <code>X</code> is small enough, you can use the Sieve of Eratosthenes to do it more efficiently. This is ideal for the "primes up to X" case since it maintains a memory of previously discarded primes. It does so by keeping a set of flags for each candidate number, all initially set to true (except for 1, of course).</p> <p>Then you take the first true value (2), output that as a prime, and then set the flags for all multiples of that to false.</p> <p>Then carry on with:</p> <ul> <li>3;</li> <li>5 (since 4 was a multiple of 2);</li> <li>7 (since 6 was a multiple of 2 and 3);</li> <li>11 (since 8 and 10 were multiples of 2 and 9 was a multiple of 3);</li> <li>13 (since 12 was a multiple of 2);</li> <li>17 (since 14 and 16 were multiples of 2 and 15 was a multiple of 3 and 5);</li> <li>and so on.</li> </ul> <p>Pseudo-code would be similar to:</p> <pre><code>def showPrimesUpTo (num): // create array of all true values array isPrime[2..num] = true // start with 2 and go until finished currNum = 2 while currNum &lt;= num: // if prime, output it if isPrime[currNum]: output currNum // also flag all multiples as nonprime clearNum = currNum * 2 while clearNum &lt;= num: isprime[clearNum] = false clearNum = clearNum + currNum // advance to next candidate currNum = currNum + 1 </code></pre> <p>Otherwise, you can do trial division as per your suggestion. The basic idea is to check each number from 2 up to the square root of your target number to see if it's a multiple. In pseudo-code, that would be something like:</p> <pre><code>def isPrime (num): // val is the value to check for factor val = 2 // only need to check so far while val * val &lt;= num: // check if an exact multiple if int (num / val) * val == num: return false // no, carry on val = val + 1 // if no factors found, it is a prime return true </code></pre> <p>The reason you only need to check up to the square root is because, if you find a factor above there, you would have already found the corresponding factor <em>below</em> the square root.</p> <p>For example, <code>3 x 17</code> is <code>51</code>. If you're checking the numbers from 2 through 50 to see if <code>51</code> is prime, you'll find <code>3</code> first, meaning you never need to check <code>17</code>.</p>
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