106
Activity Set 4.2
Chapter 4
Number Theory
MODELS FOR GREATEST COMMON
AND LEAST COMMON MULTIPLE
FACTOR
PURPOSE
To use a linear model to illustrate the concepts of greatest common factor and least common
multiple and show how they are related.
MATERIALS
No supplementary materials are needed.
INTRODUCTION
Greatest common factor (GCF) and least common multiple (LCM) are important concepts that
occur frequently in mathematics. The GCF of two numbers is usually introduced by listing all the
fact rs of two numbers, identifying the common factors, and then choosing the greatest of the
common 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, identifying 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. The
GCF 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 will
fit. This will be determined by placing copies of each rod end to end.
C:===:JIIIC===:JI==:=JI ==:=JI ==:=J
CI
.
--
CI
CI
~-
Activity Set 4.2
Greatest
Models for Greatest Common Factor and Least Common Multiple
107
Common Factor
1. Here are rods of length 36 units and 54 units. Both rods can be cut evenly into pieces with a
common 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. Mark
those 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 is
called 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 and
54 is abbreviated as GCF(36, 54).
2. Determine the greatest common factor of each pair of numbers below by indicating how you would
cut the rods into common pieces of greatest length. Record your answer next to the diagram.
*b.18~1r
251
=
~~~~=::~~~~=::~"rr""
I
GCF (18, 25)
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 you
would 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
IIIII
8 1L...-I--JL.......L.-J--'--1--L....J1
•••
12rl~-r~-r~-r~~1
~I~I~I
~I~I~I
~~
121~~~1
GCF (8,12)
=
4. Fbr each pair of rods shown below, determine the GCF of the shorter rod and the difference
rod. Indicate how you would cut the difference rod and the smaller rod on the diagrams.
Difference
A
*a.
40~1
24~1
~~~~~~~~~~~~~~~~I
~~~~~~~~~~~~~~~~~I
~I~~~~IJI-LI~IJI-L~~I
I
Difference
r-------~A~------~,
b.42~1 ~~~~~~~~~~~~~~~~~I_I~~_I~I~IJI~I~~~I~I
28~1
~~~~~~~~-L~~~~~~~~-L~I
Difference
A
I
c.
~I
~~~~~~~~I
121~~-L~~~-L~~1
21
\
~I~I-LI
~I~I-LI
~I~I~I
*5. You may have noticed in activities 1-4 that the longer rod is always cut at a point coinciding
with the end of the shorter rod. So, the GCF of two numbers can be determined by comparing the difference rod and the smaller rod. Use your results from activity 4 to determine the
following.
GCF(40, 24)
=
GCF( 42, 28)
=
GCF(21, 12)
=
6. The preceding activity suggests that the GCF oftwo numbers can be determined by computing the difference of the two numbers and then finding the GCF of the difference and the
smaller of the two numbers. If the GCF of the smaller numbers is not apparent, this method
of 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
Use this difference
each step.
a. GCF(l44, 27)
.,:111
method to find the GCF of the following pairs of num,
=
*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 of
length 3 are arranged end to end alongside a similar arrangement of rods of length 5, the
distances 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. This
least common multiple is written LCM(3, 5) = 15.
3
[II]
5
I I I I
I I I
I I I
I I I
I I I
15
30
Find the LCM of each of the following pairs of numbers by drawing the minimum number
of 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 figure
below shows the numbers 6 and 15; GCF(6, 15) = 3 is indicated by marks on the rods. The
second figure shows that LCM(6, 15) = 30.
6
15
I I I I I I I
GCF
I I I I I I I I I I I I I I I I
(6 15) = 3
'
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 the
rod diagrams in activity 7 to complete the following table. Look for a relationship involving the GCF, LCM, and the product of the two numbers.
A
B
GCF (A, B)
LCM (A, B)
(1 )
6
15
3
30
(2)
8
12
*(3)
14
21
(4)
5
7
*(5)
8
10
f
AxB
90
b. Based on your observations from the table in part a, write a brief set of directions for
finding 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 use
the 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
111
Models for Greatest Common Factor and Least Common Multiple
*c. GCF(140, 350)
=
LCM(140, 350)
d. GCF(135, 42)
LCM(l35,42)
=
=
=
JUST FOR FUN
STAR POLYGONS
Star polygons are often constructed to provide decorative
and artistic patterns. The star polygon pictured here was
formed from colored yarn around a circle of 16 equally
spaced tacks on a piece of plywood. Starting at the red tack
in the lower left and moving in a clockwise direction, the
yarn goes a step of 5 to the next red tack and then another
step of 5 to a third red tack. This procedure continues until
the yarn gets back to its starting point. In the following activities, star polygons are analyzed by using the concepts of
factor, multiple, greatest common factor, and least common
multiple."
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 of
3 spaces to point q. Three spaces from q is point r. Through
this process we eventually come back to point p, after having
hit all 14 points. The resulting figure is a star polygon.
In general, for whole numbers nand s, where It 2:: 3 and
s < It, star (n, s) denotes a star polygon with n points and
steps of s.
11st step of 31
q
r
Star (14, 3)
4A. B. Bennett, Jr., "Star Pattern ," Arithmetic Teacher 25 (January 1978): 12-14.
12d step of 31