How Fundamental Is It?

Melfried OlsonUniversity of Wyoming

Laramie, Wyoming 82071

The fundamental theorem of arithmetic�how often do we use it to doanything but find the least common multiple and greatest common divi-sor of two positive integers? Following is an activity whose explanationrests upon the fundamental theorem of arithmetic; however, this fact is

not readily noticeable.Consider the 4 x 4 matrix below (figure 1). Select one number from

the matrix, and having selected it, eliminate all other numbers in thesame row and column as the number chosen. Continue in this manneruntil no further selection can be made�you should have selected fournumbers. Find the product of these numbers.

45 165 90 15063 231 126 21042 154 84 14012 44 24 40

FIGURE 1

Repeat the process again. Is your second product the same as the first?Repeat it again�do you get the same result? This leads to a series ofquestions I pose for my students:

1. First, can you make another 4x4 table that works in a similarmanner? After a few minutes of trying, someone reports, "Make all 16entries the same," which surely answers the question. I compliment thesolution but further ask, "Can you find a non-trivial solution?" If no so-lution is forthcoming I proceed to question 2.

2. Can you find a 2 x 2 arrangement that works? A 2 x 2 arrange-ment, of course, leads to only two products from which students quicklyconclude that the product of the diagonals must be equal. We then pur-sue question 3.

3. Can you find a 3 x 3 arrangement that works? In this case there areonly six possible arrangements of products to consider, and even thoughan industrious student will try, the solution is usually not easily found.Here we suggest to consider a subset of a multiplication table, and thisusually provides a necessary clue. Suppose we consider the followingsubset of a multiplication table (figure 2) and circle 18,16 and 25. The re-sulting product is 7200, as is any other product of three numbers chosenso that there is one from each row and column.

511

512School Science and Mathematics

x | 4 5 6

3 12 15 184 16 20 245 20 25 30

FIGURE 2

I now ask the student to compare 7200 with the product of 4, 5, 6, 3, 4and 5, which solidifies the clue. This clue enables the students to generatemore 3x3 arrangements, and most will try this on 4 x 4 arrangementsand find it works satisfactorily. Similarly, this will work on any n x n ar-rangement. We now pose our last question.

4. Can you find a 4 x 4 arrangement whose resulting product is95256000. This gets the wheels turning again, and many try a method ofattack based upon the solution discussed in 3 above. Noticing that theprime factorization of 95256000 = 26 � 35 � 53 � 72, we can arrange ourmultiplication table using all factors of 95256000 once and only once(figure 3), to arrive at a solution, which of course is not unique.

2-32-2 �

2-

5537

= 10= 45= 6= 14

3 � 7=21

210945126294

21�3=12

12054072168

2

20901228

5

502253070

FIGURE 3

We can now see that, indeed, the fundamental theorem was essentialto our solution. Of course, we reinforced some problem solving conceptsof reducing to simpler cases and using previous ideas that are analogousto this problem. Finally, this provides a different slant to the two tradi-tional uses of the fundamental theorem.

SEVERE FLOODING LIKELY TO OCCURIN NORTHWEST

The National Oceanic and Atmospheric Administration (NOAA) said thatthere is a risk of severe flooding from four Northwest rivers clogged by mudflowfrom the Mount St. Helens eruption.The volcanic activity also simultaneously impaired the agency’s National

Weather Service’s (NWS) ability to forecast flood conditions. A system of auto-matic stream gauges that the NWS uses to measure river and rainfall activitywere lost after the eruption. The Weather Service is now giving top priority to re-placing the system.The new system will allow the National Weather Service to double the warning

time it can provide heavily populated areas of flooding due to lava flow or rain-fall.