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* [https://quizlet.com/Liming_Yu6/folders/math-4-recognizing-prime-and-composite-numbers/sets Math 4: Recognizing prime and composite numbers] Quizlet 資料夾
* [https://docs.google.com/spreadsheets/d/16XRvnwi4NzapP5ZbE4XfrgeeRacm9z93s-vvrlhvg8A/edit?pli=1#gid=1296446943 Math 4: Recognizing prime and composite numbers] Google sheet 
* [https://docs.google.com/document/d/1MYsC4TYmgu1oWBvaP5g9Ss_A_MYOcCxYR6STvzZDG0c/edit?pli=1#heading=h.tmfe77ahk12a Math 4: Recognizing prime and composite numbers] Google Doc

===本文===
<p class='english'><span class='englishVerse'>1</span>Determine whether the following numbers are prime, composite or neither.</p>
<p class='chinese'><span class='chineseVerse'>1</span>請確定以下各數字為質數、合數、或兩者皆非。</p>
<p class='english'><span class='englishVerse'>2</span>So just as a bit of review, a prime number is a natural number, so one of the counting numbers 1, 2, 3, 4, 5, 6, so on and so forth - that has exactly two factors.</p>
<p class='chinese'><span class='chineseVerse'>2</span>稍稍回顧一下，質數是自然數，是計數數字 1、2、3、4、5、6... 等等之一個數字，而它恰恰有兩個因數。</p>
<p class='english'><span class='englishVerse'>3</span>So its factors are 1 and itself.</p>
<p class='chinese'><span class='chineseVerse'>3</span>即其因數為 1 及它本身。</p>
<p class='english'><span class='englishVerse'>4</span>So an example of a prime is 3.</p>
<p class='chinese'><span class='chineseVerse'>4</span>所以 3 是質數的一個例子。</p>
<p class='english'><span class='englishVerse'>5</span>There's only two natural numbers that are divisible into 3 - 1 and 3.</p>
<p class='chinese'><span class='chineseVerse'>5</span>只有兩個自然數可以整除 3 即 1 和 3。</p>
<p class='english'><span class='englishVerse'>6</span>Or another way to think about it is, the only way to get 3 as a product of other natural numbers is 1 times 3.</p>
<p class='chinese'><span class='chineseVerse'>6</span>或者換一個方式思考，將 3 作為其他自然數的乘積之唯一方法是 1 乘以 3。</p>
<p class='english'><span class='englishVerse'>7</span>So it only has 1 and itself.</p>
<p class='chinese'><span class='chineseVerse'>7</span>因此它只有 1 和 它自己。</p>
<p class='english'><span class='englishVerse'>8</span>A composite number is a natural number that has more than just 1 and itself as factors.</p>
<p class='chinese'><span class='chineseVerse'>8</span>一個合數是一個自然數，它不僅僅具有 1 和它本身為其因數。</p>
<p class='english'><span class='englishVerse'>9</span>And we'll see examples of that and neither -- we’ll see an interesting case of that in this problem.</p>
<p class='chinese'><span class='chineseVerse'>9</span>我們將看到那些(合數)的例子，以及兩者皆非--我們將在本題目中看到一個有趣的例子。</p>
<p class='english'><span class='englishVerse'>10</span>So first let's think about 24.</p>
<p class='chinese'><span class='chineseVerse'>10</span>好，首先讓我們考慮 24。</p>
<p class='english'><span class='englishVerse'>11</span>So let’s think about all of the -- I guess you could think of it as the natural numbers or the whole numbers,  although 0 is also included in whole numbers.</p>
<p class='chinese'><span class='chineseVerse'>11</span>因此，讓我們考慮所有 -- 我猜您可能將其 (24) 代表所有自然數或整數，儘管整數中也包含 0。</p>
<p class='english'><span class='englishVerse'>12</span>Let’s think of all of the natural counting numbers that we can actually divide into 24 without having any remainder.</p>
<p class='chinese'><span class='chineseVerse'>12</span>讓我們考慮所有那些我們可以用來整除 24 而沒有餘數的自然數。</p>
<p class='english'><span class='englishVerse'>13</span>We’d consider those the factors.</p>
<p class='chinese'><span class='chineseVerse'>13</span>我們要評估它們是否為因數。</p>
<p class='english'><span class='englishVerse'>14</span>Well, clearly it is divisible by 1 and 24.</p>
<p class='chinese'><span class='chineseVerse'>14</span>好吧！顯然它(24)可以被 1 和 24 整除。</p>
<p class='english'><span class='englishVerse'>15</span>In fact, 1 times 24 is equal to 24.</p>
<p class='chinese'><span class='chineseVerse'>15</span>事實上，1 乘以 24 等於 24。</p>
<p class='english'><span class='englishVerse'>16</span>But it's also divisible by 2.</p>
<p class='chinese'><span class='chineseVerse'>16</span>而它(24)也可以被 2 整除。</p>
<p class='english'><span class='englishVerse'>17</span>2 times 12 is 24.</p>
<p class='chinese'><span class='chineseVerse'>17</span>2 乘以 12 得 24。</p>
<p class='english'><span class='englishVerse'>18</span>So it’s also divisible by 12.</p>
<p class='chinese'><span class='chineseVerse'>18</span>因此它(24)也可以被 12 整除。</p>
<p class='english'><span class='englishVerse'>19</span>And it is also divisible by 3.</p>
<p class='chinese'><span class='chineseVerse'>19</span>而且它(24)也可以被 3 整除。</p>
<p class='english'><span class='englishVerse'>20</span>3 times 8 is also equal to 24.</p>
<p class='chinese'><span class='chineseVerse'>20</span>3 乘以 8 也等於 24。</p>
<p class='english'><span class='englishVerse'>21</span>And even at this point, we don't actually have to find all of the factors to realize it's not prime.</p>
<p class='chinese'><span class='chineseVerse'>21</span>而甚至在此時，我們實際上並不需要找出所有因數來確定它(24)不是質數。</p>
<p class='english'><span class='englishVerse'>22</span>It clearly has more factors than just 1 and itself.</p>
<p class='chinese'><span class='chineseVerse'>22</span>它顯然有 1 和 24 之外更多的因數。</p>
<p class='english'><span class='englishVerse'>23</span>So then it is clearly going to be composite.</p>
<p class='chinese'><span class='chineseVerse'>23</span>因此它(24)顯然將會是一個合數。</p>
<p class='english'><span class='englishVerse'>24</span>This is going to be composite.</p>
<p class='chinese'><span class='chineseVerse'>24</span>這將會是合數。</p>
<p class='english'><span class='englishVerse'>25</span>Now, let’s just finish factoring it just since we started it.</p>
<p class='chinese'><span class='chineseVerse'>25</span>現在，既然我們已經開始了，就讓我們完成它(24)的因數分解。</p>
<p class='english'><span class='englishVerse'>26</span>It’s also divisible by 4.</p>
<p class='chinese'><span class='chineseVerse'>26</span>它(24)也可以被 4 整除。</p>
<p class='english'><span class='englishVerse'>27</span>And 4 times 6 -- had just enough space to do that.</p>
<p class='chinese'><span class='chineseVerse'>27</span>還有 4 乘以 6 -- 正好有足够的空間可以做這個。</p>
<p class='english'><span class='englishVerse'>28</span>4 times 6 is also 24.</p>
<p class='chinese'><span class='chineseVerse'>28</span>4 乘以 6 也是 24。</p>
<p class='english'><span class='englishVerse'>29</span>So these are all of the factors of 24, clearly more than just 1 in 24.</p>
<p class='chinese'><span class='chineseVerse'>29</span>因此這些都是 24 的因數，顯然的不只是 1 和 24。</p>
<p class='english'><span class='englishVerse'>30</span>Now let’s think about 2.</p>
<p class='chinese'><span class='chineseVerse'>30</span>現在讓我們來看看 2。</p>
<p class='english'><span class='englishVerse'>31</span>Well, the non-zero whole numbers that are divisible into 2, well, 1 times 2 definitely works, 1 and 2.</p>
<p class='chinese'><span class='chineseVerse'>31</span>是的，可以整除 2 的非零整數，是的，1 乘以 2 必定可以，1 和 2。</p>
<p class='english'><span class='englishVerse'>32</span>But there really aren't any others that are divisible into 2.</p>
<p class='chinese'><span class='chineseVerse'>32</span>可是真的沒有其他的數可以整除 2。</p>
<p class='english'><span class='englishVerse'>33</span>And so it only has two factors, 1 and itself, and that's the definition of a prime number.</p>
<p class='chinese'><span class='chineseVerse'>33</span>因此它(2)只有兩個因數，即 1 和自己，而這就是質數的定義。</p>
<p class='english'><span class='englishVerse'>34</span>So 2 is prime.</p>
<p class='chinese'><span class='chineseVerse'>34</span>所以 2 是質數。</p>
<p class='english'><span class='englishVerse'>35</span>And 2 is interesting because it is the only even prime number.</p>
<p class='chinese'><span class='chineseVerse'>35</span>另外， 2 很有意思，因為它(2)是唯一的偶質數。</p>
<p class='english'><span class='englishVerse'>36</span>And that might be common sense to you.</p>
<p class='chinese'><span class='chineseVerse'>36</span>而這可能對你而言是常識。</p>
<p class='english'><span class='englishVerse'>37</span>Because by definition, an even number is divisible by 2.</p>
<p class='chinese'><span class='chineseVerse'>37</span>因為根據定義，偶數就是可以被 2 整除的數。</p>
<p class='english'><span class='englishVerse'>38</span>So 2 is clearly divisible by 2.</p>
<p class='chinese'><span class='chineseVerse'>38</span>所以 2 顯然的可以被 2 整除。</p>
<p class='english'><span class='englishVerse'>39</span>That's what makes it even.</p>
<p class='chinese'><span class='chineseVerse'>39</span>這使得它(2)成為偶數。</p>
<p class='english'><span class='englishVerse'>40</span>But it’s only divisible by 2 and 1.</p>
<p class='chinese'><span class='chineseVerse'>40</span>而它(2)也只能被 2 和 1 整除。</p>
<p class='english'><span class='englishVerse'>41</span>So that's what makes it prime.</p>
<p class='chinese'><span class='chineseVerse'>41</span>這使得它(2)因而成為質數。</p>
<p class='english'><span class='englishVerse'>42</span>But anything that’s even is going to be divisible by 1, itself and 2.</p>
<p class='chinese'><span class='chineseVerse'>42</span>另外，任何的偶數都可以被 1，它自己，和 2 整除。</p>
<p class='english'><span class='englishVerse'>43</span>Any other number that is even is going to be visible by 1, itself, and 2.</p>
<p class='chinese'><span class='chineseVerse'>43</span>任何其他的偶數都可以被  1，它自己，和 2 整除。</p>
<p class='english'><span class='englishVerse'>44</span>So by definition, it’s going to have 1 and itself and something else.</p>
<p class='chinese'><span class='chineseVerse'>44</span>所以根據定義，它(任何其他的偶數)將有   1，它自己和其他數(為因數)。</p>
<p class='english'><span class='englishVerse'>45</span>So it’s going to be composite.</p>
<p class='chinese'><span class='chineseVerse'>45</span>因此它(任何其他的偶數)將會是合數。</p>
<p class='english'><span class='englishVerse'>46</span>So 2 is prime.</p>
<p class='chinese'><span class='chineseVerse'>46</span>故 2 是質數。</p>
<p class='english'><span class='englishVerse'>47</span>Every other even number other than 2 is composite.</p>
<p class='chinese'><span class='chineseVerse'>47</span>其他不是 2 的偶數都是合數。</p>
<p class='english'><span class='englishVerse'>48</span>Now, here is an interesting case.</p>
<p class='chinese'><span class='chineseVerse'>48</span>再來，這兒有一個有趣的例子。</p>
<p class='english'><span class='englishVerse'>49</span>1--1 is only divisible by 1.</p>
<p class='chinese'><span class='chineseVerse'>49</span>1-- 1 只能被 1 整除。</p>
<p class='english'><span class='englishVerse'>50</span>So it is not prime, technically, because it only has 1 as a factor.</p>
<p class='chinese'><span class='chineseVerse'>50</span>所以它(1)不是質數，技術上來說，因為它(1)只有一個因數。</p>
<p class='english'><span class='englishVerse'>51</span>It does not have two factors.</p>
<p class='chinese'><span class='chineseVerse'>51</span>它(1)沒有兩個因數。</p>
<p class='english'><span class='englishVerse'>52</span>1 is itself.</p>
<p class='chinese'><span class='chineseVerse'>52</span>1 是它自己。</p>
<p class='english'><span class='englishVerse'>53</span>But in order to be prime, you have to have exactly two factors.</p>
<p class='chinese'><span class='chineseVerse'>53</span>但要成為質數，你必須要恰有兩個因數。</p>
<p class='english'><span class='englishVerse'>54</span>1 has only one factor.</p>
<p class='chinese'><span class='chineseVerse'>54</span>1 只有一個因數。</p>
<p class='english'><span class='englishVerse'>55</span>In order to be composite you have to have more than two factors.</p>
<p class='chinese'><span class='chineseVerse'>55</span>但要成為合數，你必須要有兩個以上的因數。</p>
<p class='english'><span class='englishVerse'>56</span>You have to have 1, yourself, and some other numbers.</p>
<p class='chinese'><span class='chineseVerse'>56</span>你必須要有 1，你自己，和一些其他的數。</p>
<p class='english'><span class='englishVerse'>57</span>So it's not composite.</p>
<p class='chinese'><span class='chineseVerse'>57</span>因此它(1)不是合數。</p>
<p class='english'><span class='englishVerse'>58</span>So 1 is neither prime nor composite.</p>
<p class='chinese'><span class='chineseVerse'>58</span>所以 1 既不是質數也不是合數。</p>
<p class='english'><span class='englishVerse'>59</span>And then finally we get to 17.</p>
<p class='chinese'><span class='chineseVerse'>59</span>接下來終於我們來到 17。</p>
<p class='english'><span class='englishVerse'>60</span>17 is divisible by 1 and 17.</p>
<p class='chinese'><span class='chineseVerse'>60</span>17 可以被 1 和 17 整除。</p>
<p class='english'><span class='englishVerse'>61</span>It’s not divisible by 2, not divisible by 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, or 16.</p>
<p class='chinese'><span class='chineseVerse'>61</span>它(17)可能被 2 整除，不能被 3,4…, 16 整除。</p>
<p class='english'><span class='englishVerse'>62</span>So it has exactly two factors -- 1 and itself.</p>
<p class='chinese'><span class='chineseVerse'>62</span>所以它(17)恰有兩個因數--即 1 和它自己。</p>
<p class='english'><span class='englishVerse'>63</span>So 17 is once again -- 17 is prime.</p>
<p class='chinese'><span class='chineseVerse'>63</span>故 17 又是 -- 17 是質數。</p>