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Activity Set 4.2 MODELS FOR GREATEST COMMON FACTOR AND ...

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106 Chapter 4 Number Theory<br />

<strong>Activity</strong> <strong>Set</strong> <strong>4.2</strong> <strong>MODELS</strong> <strong>FOR</strong> <strong>GREATEST</strong> <strong>COMMON</strong> <strong>FACTOR</strong><br />

<strong>AND</strong> LEAST <strong>COMMON</strong> MULTIPLE<br />

PURPOSE<br />

To use a linear model to illustrate the concepts of greatest common factor and least common<br />

multiple and show how they are related.<br />

MATERIALS<br />

No supplementary materials are needed.<br />

INTRODUCTION<br />

Greatest common factor (GCF) and least common multiple (LCM) are important concepts that<br />

occur frequently in mathematics. The GCF of two numbers is usually introduced by listing all the<br />

fact rs of two numbers, identifying the common factors, and then choosing the greatest of the<br />

common factors.<br />

Factors of 12: L~, ~,4,@12<br />

Factors of IS: i, i, 3,@)9, IS<br />

The LCM of two numbers is often introduced by listing multiples of each number, identifying<br />

common multiples, and choosing the least of the common multiples.<br />

Multiples of 12: 12, 24,®4S, 60, .72, S4, 96, I.9S, 120, 132, .1:~-4,...<br />

Multiples of IS: IS,@S4, 7~, 90, His, 126, 1M', 162, ISO, 19S, ...<br />

In this activity set, the concepts of GCF and LCM will be visually represented by rods. The<br />

GCF will be viewed as the greatest common length into which two (or more) rods can be cut.<br />

The LCM will be viewed as the shortest common length into which two (or more) rods will<br />

fit. This will be determined by placing copies of each rod end to end.<br />

C:===:JIIIC===:JI<br />

CI ==:=JI CI<br />

==:=JI<br />

·.<br />

CI<br />

==:=J<br />

~-


1 1<br />

<strong>Activity</strong> <strong>Set</strong> <strong>4.2</strong> Models for Greatest Common Factor and Least Common Multiple 107<br />

Greatest Common Factor<br />

1. Here are rods of length 36 units and 54 units. Both rods can be cut evenly into pieces with a<br />

common length of 6 units, since 6 is a factor of 36 and 6 is a factor of 54.<br />

a. There are five other ways to cut both rods evenly into pieces of common length. Mark<br />

those on the following five pairs of rods.<br />

*b. Because rods of length 36 and 54 can both be cut evenly into pieces of length 6, 6 is<br />

called a common factor of 36 and 54. List the other common factors of 36 and 54.<br />

Common factors of 36 and 54: 6, __ , __ , __ , __ , __<br />

c. Circle the greatest common factor' of 36 and 54. The greatest common factor of 36 and<br />

54 is abbreviated as GCF(36, 54).<br />

2. Determine the greatest common factor of each pair of numbers below by indicating how you would<br />

cut the rods into common pieces of greatest length. Record your answer next to the diagram.<br />

*b.18~1 r~~~~=::~~~~=::~"rr"" GCF (18, 25) =<br />

251 I<br />

3Also commonly called the greatest common divisor or GCD.


108 Chapter 4 Number Theory<br />

3. a. Determine the greatest common factor of the numbers 20 and 12 by indicating how you<br />

would cut the two rods below into pieces of greatest common length.<br />

b. The amount by which one rod exceeds the other is represented by the difference rod.<br />

Determine the GCF of the difference rod and the shorter rod using the diagram below.<br />

Difference<br />

~<br />

20 I I I I I I I I II<br />

12rl~-r~-r~-r~~1<br />

••• 8 1L...-I--JL.......L.-J--'--1--L....J1<br />

121~~~1 ~I~I~I<br />

GCF (8,12)=<br />

~I~I~I ~~<br />

4. Fbr each pair of rods shown below, determine the GCF of the shorter rod and the difference<br />

rod. Indicate how you would cut the difference rod and the smaller rod on the diagrams.<br />

Difference<br />

A<br />

*a. 40~1~~~~~~~~~~~~~~~~I ~I~~~~IJI-LI~IJI-L~~I<br />

24~1~~~~~~~~~~~~~~~~~I<br />

Difference<br />

r-------~A~------~,<br />

b.42~1 ~~~~~~~~~~~~~~~~~I_I~~_I~I~IJI~I~~~I~I<br />

28~1~~~~~~~~-L~~~~~~~~-L~I<br />

Difference<br />

A<br />

I \<br />

c. 21 ~I~~~~~~~~I ~I~I-LI ~I~I-LI ~I~I~I<br />

121~~-L~~~-L~~1<br />

*5. You may have noticed in activities 1-4 that the longer rod is always cut at a point coinciding<br />

with the end of the shorter rod. So, the GCF of two numbers can be determined by comparing<br />

the difference rod and the smaller rod. Use your results from activity 4 to determine the<br />

following.<br />

GCF(40, 24) = GCF( 42, 28) = GCF(21, 12) =<br />

6. The preceding activity suggests that the GCF oftwo numbers can be determined by computing<br />

the difference of the two numbers and then finding the GCF of the difference and the<br />

smaller of the two numbers. If the GCF of the smaller numbers is not apparent, this method<br />

of taking differences can be continued. For example,<br />

GCF(l98, 126) = GCF(72, 126) = GCF(72, 54) = GCF(18, 54)<br />

= GCF(18, 36) = GCF(18, 18) = 18


<strong>Activity</strong> <strong>Set</strong> <strong>4.2</strong> Models for Greatest Common Factor and Least Common Multiple .,:111<br />

Use this difference method to find the GCF of the following pairs of num,<br />

each step.<br />

a. GCF(l44, 27) =<br />

*b. GCF(280, 168) =<br />

c. GCF(714,420) =<br />

*d. GCF(306, 187) =<br />

Least Common Multiple<br />

7. In the following diagram, the numbers 3 and 5 are represented by rods. When the rods of<br />

length 3 are arranged end to end alongside a similar arrangement of rods of length 5, the<br />

distances at which the ends evenly match are common multiples of 3 and 5 (15,30,45, etc.).<br />

The least distance at which they match, 15, is the least common multiple of 3 and 5. This<br />

least common multiple is written LCM(3, 5) = 15.<br />

3 [II]<br />

5 I I I I<br />

I I I I I I<br />

I I I I I I<br />

15 30<br />

Find the LCM of each of the following pairs of numbers by drawing the minimum number<br />

of end-to-end rods of each length needed to make both rows the same length.<br />

a. 8~1~~~~~~~~~~~~~~~~~~~~~~~~~~~<br />

12~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br />

*b.14~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br />

21~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br />

LCM (8, 12) = __<br />

LCM (14, 21) = __<br />

c. 5~1~~~~~~~~~~~~~~~~~~~~~~~~~~~<br />

7~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~<br />

LCM (5, 7) = __<br />

LCM (8, 10) = __


110 Chapter 4 Number Theory<br />

8. There is a relationship between the GCF and the LCM of two numbers. The first figure<br />

below shows the numbers 6 and 15; GCF(6, 15) = 3 is indicated by marks on the rods. The<br />

second figure shows that LCM(6, 15) = 30.<br />

6 I I I I I I I GCF (6 15) = 3<br />

15 I I I I I I I I I I I I I I I I '<br />

a. GCF(6, 15) = 3 and 3 divides the 6-unit rod into 2 parts and the IS-unit rod into 5 parts.<br />

Notice that 2 rods of length 15 or 5 rods of length 6 equal 30, the LCM(6, 15). Use the<br />

rod diagrams in activity 7 to complete the following table. Look for a relationship involving<br />

the GCF, LCM, and the product of the two numbers.<br />

(1 )<br />

f (2)<br />

*(3)<br />

(4)<br />

*(5)<br />

A B GCF (A, B) LCM (A, B) AxB<br />

6<br />

8<br />

14<br />

5<br />

8<br />

15 3 30 90<br />

12<br />

21<br />

7<br />

10<br />

b. Based on your observations from the table in part a, write a brief set of directions for<br />

finding the LCM of two numbers once you have determined the GCF.<br />

9. For each of the following pairs of numbers, first compute the GCF of the pair and then use<br />

the relationship from activity 8b to compute the LCM.<br />

*a. GCF(9, 15) =<br />

LCM(9, 15) =<br />

b. GCF(8, 18) =<br />

LCM(8, 18) =


<strong>Activity</strong> <strong>Set</strong> <strong>4.2</strong> Models for Greatest Common Factor and Least Common Multiple 111<br />

*c. GCF(140, 350) =<br />

LCM(140, 350) =<br />

d. GCF(135, 42) =<br />

LCM(l35,42) =<br />

JUST <strong>FOR</strong> FUN<br />

STAR POLYGONS Star polygons can be constructed by taking steps of a<br />

Star polygons are often constructed to provide decorative<br />

and artistic patterns. The star polygon pictured here was<br />

formed from colored yarn around a circle of 16 equally<br />

spaced tacks on a piece of plywood. Starting at the red tack<br />

in the lower left and moving in a clockwise direction, the<br />

yarn goes a step of 5 to the next red tack and then another<br />

step of 5 to a third red tack. This procedure continues until<br />

the yarn gets back to its starting point. In the following activities,<br />

star polygons are analyzed by using the concepts of<br />

factor, multiple, greatest common factor, and least common<br />

multiple."<br />

4A. B. Bennett, Jr., "Star Pattern ," Arithmetic Teacher 25 (January 1978): 12-14.<br />

given size around a circle of points. The following star (14, 3)<br />

was constructed by beginning at point p and taking a step of<br />

3 spaces to point q. Three spaces from q is point r. Through<br />

this process we eventually come back to point p, after having<br />

hit all 14 points. The resulting figure is a star polygon.<br />

In general, for whole numbers nand s, where It 2:: 3 and<br />

s < It, star (n, s) denotes a star polygon with n points and<br />

steps of s.<br />

11st step of 31 q<br />

Star (14, 3)<br />

r 12d step of 31

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