The Investigation poses questions to generate interest in various
mathematical topics from the text and encourages students to formulate and investigate their
own conjectures. One use of the investigations is for term papers in which students report on
their conjectures and the patterns they find.
Click on the Read Me file below to open the investigation in a Word file:
 Read Me - Factorizations Instructions (Word Format)
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The Math Investigator is a data collection software program that may be used to collect data for the investigation. You may type answers onto the Word file or copy
and paste in data from the Investigator. Click here to launch the Factorizations Investigator
Math Investigator 4.2FACTORIZATIONS on the Math Investigator determines if a number is prime or composite. If a number is composite, it prints all its factors, the number of factors, and its prime factorization. The numbers 1, 2, 4, and 6 have 1, 2, 3, and 4 factors, respectively: 1 has only 1 as a factor; 2 has 1 and 2 as factors; 4 has 1, 2, and 4 as factors; and 6 has 1, 2, 3, and 6 as factors. These factors are illustrated by the rectangles shown here.  <![CDATA[<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0072532947/78570/4_2.jpg','popWin', 'width=630,height=348,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (29.0K)</a>]]>
Starting Points for Investigations- Print the numbers from 1 to 100 to determine their numbers of factors. Use colored pens to mark each type of number having; exactly 3 factors; exactly 4 factors; and exactly 5 factors. Look for patterns in these numbers and their prime factorizations. Form some conjectures and cite evidence which supports them.
- The smallest numbers that have 2, 3, and 4 factors are shown in the following table. Find the smallest numbers that have 5, 6, 7, 8, 9, and 10 factors and record them and their prime factorizations. Look for patterns and predict the smallest numbers that will have 11 factors, 12 factors, and 13 factors. Test your predictions and write some conjectures based on your observations.
| Number of Factors | Number | Prime Factorization | | 2 | 2 | | | 3 | 4 | 2 x 2 | | 4 | 6 | 2 x 3 | | 5 | | | | 6 | | | | 7 | | | | 8 | | | | 9 | | | | 10 | | |
- Extend the preceding table by printing intervals of 50 numbers from 101 to 150; 151 to 200; etc. Revise or extend your conjectures regarding the smallest number with a given number of factors.
- (Optional) Describe a method for finding the smallest possible number with a given number of factors. For example, what is the smallest number that has exactly 200 factors?
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