Question 987822
Let's make a table. The first column represents the number X. For example in row 8, we have X = 8. We're finding the divisors of the number X = 8. The same applies for the other rows as well. 


The second column represents the number of divisors (eg: the number 7 has 2 divisors, so a 2 will go in the second column in row 7). 


The third column represents the actual list of divisors so you can count yourself to check the value in the second column.


We'll stop once we have the second column have a "7" in it



<table border=1><tr><th>X</th><th>number of divisors for number X</th><th>list of divisors for number X</th></tr><tr><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>2</td><td>1,2</td></tr><tr><td>3</td><td>2</td><td>1,3</td></tr><tr><td>4</td><td>3</td><td>1,2,4</td></tr><tr><td>5</td><td>2</td><td>1,5</td></tr><tr><td>6</td><td>4</td><td>1,2,3,6</td></tr><tr><td>7</td><td>2</td><td>1,7</td></tr><tr><td>8</td><td>4</td><td>1,2,4,8</td></tr><tr><td>9</td><td>3</td><td>1,3,9</td></tr><tr><td>10</td><td>4</td><td>1,2,5,10</td></tr><tr><td>11</td><td>2</td><td>1,11</td></tr><tr><td>12</td><td>6</td><td>1,2,3,4,6,12</td></tr><tr><td>13</td><td>2</td><td>1,13</td></tr><tr><td>14</td><td>4</td><td>1,2,7,14</td></tr><tr><td>15</td><td>4</td><td>1,3,5,15</td></tr><tr><td>16</td><td>5</td><td>1,2,4,8,16</td></tr><tr><td>17</td><td>2</td><td>1,17</td></tr><tr><td>18</td><td>6</td><td>1,2,3,6,9,18</td></tr><tr><td>19</td><td>2</td><td>1,19</td></tr><tr><td>20</td><td>6</td><td>1,2,4,5,10,20</td></tr><tr><td>21</td><td>4</td><td>1,3,7,21</td></tr><tr><td>22</td><td>4</td><td>1,2,11,22</td></tr><tr><td>23</td><td>2</td><td>1,23</td></tr><tr><td>24</td><td>8</td><td>1,2,3,4,6,8,12,24</td></tr><tr><td>25</td><td>3</td><td>1,5,25</td></tr><tr><td>26</td><td>4</td><td>1,2,13,26</td></tr><tr><td>27</td><td>4</td><td>1,3,9,27</td></tr><tr><td>28</td><td>6</td><td>1,2,4,7,14,28</td></tr><tr><td>29</td><td>2</td><td>1,29</td></tr><tr><td>30</td><td>8</td><td>1,2,3,5,6,10,15,30</td></tr><tr><td>31</td><td>2</td><td>1,31</td></tr><tr><td>32</td><td>6</td><td>1,2,4,8,16,32</td></tr><tr><td>33</td><td>4</td><td>1,3,11,33</td></tr><tr><td>34</td><td>4</td><td>1,2,17,34</td></tr><tr><td>35</td><td>4</td><td>1,5,7,35</td></tr><tr><td>36</td><td>9</td><td>1,2,3,4,6,9,12,18,36</td></tr><tr><td>37</td><td>2</td><td>1,37</td></tr><tr><td>38</td><td>4</td><td>1,2,19,38</td></tr><tr><td>39</td><td>4</td><td>1,3,13,39</td></tr><tr><td>40</td><td>8</td><td>1,2,4,5,8,10,20,40</td></tr><tr><td>41</td><td>2</td><td>1,41</td></tr><tr><td>42</td><td>8</td><td>1,2,3,6,7,14,21,42</td></tr><tr><td>43</td><td>2</td><td>1,43</td></tr><tr><td>44</td><td>6</td><td>1,2,4,11,22,44</td></tr><tr><td>45</td><td>6</td><td>1,3,5,9,15,45</td></tr><tr><td>46</td><td>4</td><td>1,2,23,46</td></tr><tr><td>47</td><td>2</td><td>1,47</td></tr><tr><td>48</td><td>10</td><td>1,2,3,4,6,8,12,16,24,48</td></tr><tr><td>49</td><td>3</td><td>1,7,49</td></tr><tr><td>50</td><td>6</td><td>1,2,5,10,25,50</td></tr><tr><td>51</td><td>4</td><td>1,3,17,51</td></tr><tr><td>52</td><td>6</td><td>1,2,4,13,26,52</td></tr><tr><td>53</td><td>2</td><td>1,53</td></tr><tr><td>54</td><td>8</td><td>1,2,3,6,9,18,27,54</td></tr><tr><td>55</td><td>4</td><td>1,5,11,55</td></tr><tr><td>56</td><td>8</td><td>1,2,4,7,8,14,28,56</td></tr><tr><td>57</td><td>4</td><td>1,3,19,57</td></tr><tr><td>58</td><td>4</td><td>1,2,29,58</td></tr><tr><td>59</td><td>2</td><td>1,59</td></tr><tr><td>60</td><td>12</td><td>1,2,3,4,5,6,10,12,15,20,30,60</td></tr><tr><td>61</td><td>2</td><td>1,61</td></tr><tr><td>62</td><td>4</td><td>1,2,31,62</td></tr><tr><td>63</td><td>6</td><td>1,3,7,9,21,63</td></tr><tr><td>64</td><td>7</td><td>1,2,4,8,16,32,64</td></tr></table>



Things to notice
1) The prime numbers only have 2 divisors. This is always true because that's what a prime number is: a number divisible by itself and 1 (and no other number).
2) The perfect squares have an odd number of divisors while any other number has an even number of divisors. So because we want 7 divisors, this means that we could restrict ourselves to just looking at the perfect squares. It makes a much nicer table that is smaller


<table border=1><tr><th>X</th><th>number of divisors for number X</th><th>list of divisors for number X</th></tr><tr><td>1</td><td>1</td><td>1</td></tr><tr><td>4</td><td>3</td><td>1,2,4</td></tr><tr><td>9</td><td>3</td><td>1,3,9</td></tr><tr><td>16</td><td>5</td><td>1,2,4,8,16</td></tr><tr><td>25</td><td>3</td><td>1,5,25</td></tr><tr><td>36</td><td>9</td><td>1,2,3,4,6,9,12,18,36</td></tr><tr><td>49</td><td>3</td><td>1,7,49</td></tr><tr><td>64</td><td>7</td><td>1,2,4,8,16,32,64</td></tr></table>



From either table, we see that the number <font color="red">64</font> is the smallest positive integer that has exactly 7 positive divisors (or factors). Those divisors are 1,2,4,8,16,32,64


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If you need more help, or if you have any questions about the problem, feel free to email me at <font color="blue"><code>jim_thompson5910@hotmail.com</code></font>