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106 Chapter 4 Number Theory

Activity Set 4.2 MODELS FOR GREATEST COMMON FACTORAND LEAST COMMON MULTIPLE

PURPOSETo use a linear model to illustrate the concepts of greatest common factor and least commonmultiple and show how they are related.

MATERIALSNo supplementary materials are needed.

INTRODUCTIONGreatest common factor (GCF) and least common multiple (LCM) are important concepts thatoccur frequently in mathematics. The GCF of two numbers is usually introduced by listing all thefact rs of two numbers, identifying the common factors, and then choosing the greatest of thecommon factors.

Factors of 12: L~, ~,4,@12

Factors of IS: i, i, 3,@)9, IS

The LCM of two numbers is often introduced by listing multiples of each number, identify-ing common multiples, and choosing the least of the common multiples.

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

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

In this activity set, the concepts of GCF and LCM will be visually represented by rods. TheGCF will be viewed as the greatest common length into which two (or more) rods can be cut.

The LCM will be viewed as the shortest common length into which two (or more) rods willfit. This will be determined by placing copies of each rod end to end.

C:===:JIIIC===:JICI ==:=JI CI ==:=JI CI ==:=J

. -- ~-

Activity Set 4.2 Models for Greatest Common Factor and Least Common Multiple 107

Greatest Common Factor

1. Here are rods of length 36 units and 54 units. Both rods can be cut evenly into pieces with acommon length of 6 units, since 6 is a factor of 36 and 6 is a factor of 54.

a. There are five other ways to cut both rods evenly into pieces of common length. Markthose on the following five pairs of rods.

1 1

*b. Because rods of length 36 and 54 can both be cut evenly into pieces of length 6, 6 iscalled a common factor of 36 and 54. List the other common factors of 36 and 54.

Common factors of 36 and 54: 6, __ , __ , __ , __ , __

c. Circle the greatest common factor' of 36 and 54. The greatest common factor of 36 and54 is abbreviated as GCF(36, 54).

2. Determine the greatest common factor of each pair of numbers below by indicating how you wouldcut the rods into common pieces of greatest length. Record your answer next to the diagram.

*b.18~1 ~~~~=::~~~~=::~"rr""r GCF (18, 25) =251 I

3Also commonly called the greatest common divisor or GCD.

108 Chapter 4 Number Theory

3. a. Determine the greatest common factor of the numbers 20 and 12 by indicating how youwould cut the two rods below into pieces of greatest common length.

b. The amount by which one rod exceeds the other is represented by the difference rod.Determine the GCF of the difference rod and the shorter rod using the diagram below.

Difference

~20 I I I I I IIII I12rl~-r~-r~-r~~1

••• 8 1L...-I--JL.......L.-J--'--1--L....J1

121~~~1 ~I~I~I~I~I~I~~GCF (8,12)=

4. Fbr each pair of rods shown below, determine the GCF of the shorter rod and the differencerod. Indicate how you would cut the difference rod and the smaller rod on the diagrams.

DifferenceA

*a. 40~1~~~~~~~~~~~~~~~~I ~I~~~~IJI-LI~IJI-L~~II24~1~~~~~~~~~~~~~~~~~I

Differencer-------~A~------~,b.42~1 ~~~~~~~~~~~~~~~~~I_I~~_I~I~IJI~I~~~I~I

28~1~~~~~~~~-L~~~~~~~~-L~I

DifferenceA

I \

c. 21 ~I~~~~~~~~I ~I~I-LI~I~I-LI~I~I~I121~~-L~~~-L~~1

*5. You may have noticed in activities 1-4 that the longer rod is always cut at a point coincidingwith the end of the shorter rod. So, the GCF of two numbers can be determined by compar-ing the difference rod and the smaller rod. Use your results from activity 4 to determine thefollowing.

GCF(40, 24) = GCF( 42, 28) = GCF(21, 12) =

6. The preceding activity suggests that the GCF oftwo numbers can be determined by comput-ing the difference of the two numbers and then finding the GCF of the difference and thesmaller of the two numbers. If the GCF of the smaller numbers is not apparent, this methodof taking differences can be continued. For example,

GCF(l98, 126) = GCF(72, 126) = GCF(72, 54) = GCF(18, 54)= GCF(18, 36) = GCF(18, 18) = 18

Activity Set 4.2 Models for Greatest Common Factor and Least Common Multiple .,:111

Use this difference method to find the GCF of the following pairs of num,each step.

a. GCF(l44, 27) =

*b. GCF(280, 168) =

c. GCF(714,420) =

*d. GCF(306, 187) =

Least Common Multiple

7. In the following diagram, the numbers 3 and 5 are represented by rods. When the rods oflength 3 are arranged end to end alongside a similar arrangement of rods of length 5, thedistances at which the ends evenly match are common multiples of 3 and 5 (15,30,45, etc.).The least distance at which they match, 15, is the least common multiple of 3 and 5. Thisleast common multiple is written LCM(3, 5) = 15.

3 [II]5 I I I I

I I I I I II I I I I I15 30

Find the LCM of each of the following pairs of numbers by drawing the minimum numberof end-to-end rods of each length needed to make both rows the same length.

a. 8~1~~~~~~~~~~~~~~~~~~~~~~~~~~~12~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~

LCM (8, 12) = __

*b.14~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~21~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~

LCM (14, 21) = __

c. 5~1~~~~~~~~~~~~~~~~~~~~~~~~~~~7~1~~~~~~~~~~~~~~~~~~~~~~~~~~~~LCM (5, 7) = __

LCM (8, 10) = __

110 Chapter 4 Number Theory

8. There is a relationship between the GCF and the LCM of two numbers. The first figurebelow shows the numbers 6 and 15; GCF(6, 15) = 3 is indicated by marks on the rods. Thesecond figure shows that LCM(6, 15) = 30.

6 I I I I I I I GCF (6 15) = 315 I I I I I I I I I I I I I I I I '

a. GCF(6, 15) = 3 and 3 divides the 6-unit rod into 2 parts and the IS-unit rod into 5 parts.Notice that 2 rods of length 15 or 5 rods of length 6 equal 30, the LCM(6, 15). Use therod diagrams in activity 7 to complete the following table. Look for a relationship involv-ing the GCF, LCM, and the product of the two numbers.

A GCF (A, B) AxBB LCM (A, B)

(1 )

f (2)

*(3)(4)

*(5)

6

8

14

5

8

15 3 30 9012

21

7

10

b. Based on your observations from the table in part a, write a brief set of directions forfinding the LCM of two numbers once you have determined the GCF.

9. For each of the following pairs of numbers, first compute the GCF of the pair and then usethe relationship from activity 8b to compute the LCM.

*a. GCF(9, 15) =

LCM(9, 15) =

b. GCF(8, 18) =

LCM(8, 18) =

Activity Set 4.2 Models for Greatest Common Factor and Least Common Multiple 111

*c. GCF(140, 350) =

LCM(140, 350) =

d. GCF(135, 42) =

LCM(l35,42) =

JUST FOR FUNSTAR POLYGONS Star polygons can be constructed by taking steps of a

given size around a circle of points. The following star (14, 3)was constructed by beginning at point p and taking a step of3 spaces to point q. Three spaces from q is point r. Throughthis process we eventually come back to point p, after havinghit all 14 points. The resulting figure is a star polygon.

In general, for whole numbers nand s, where It 2:: 3 ands < It, star (n, s) denotes a star polygon with n points andsteps of s.

11st step of 31 q

Star polygons are often constructed to provide decorativeand artistic patterns. The star polygon pictured here wasformed from colored yarn around a circle of 16 equallyspaced tacks on a piece of plywood. Starting at the red tackin the lower left and moving in a clockwise direction, theyarn goes a step of 5 to the next red tack and then anotherstep of 5 to a third red tack. This procedure continues untilthe yarn gets back to its starting point. In the following acti-vities, star polygons are analyzed by using the concepts offactor, multiple, greatest common factor, and least commonmultiple."

r 12d step of 31

Star (14, 3)

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