the greatest common factor and the least common multiple

Table of Contents
Objectives ............................................................................................................................ 4
Lesson 1: Finding the Greatest Common Factor by Continuous Division
- Warm Up! .................................................................................................................. 5
Lesson 2: Finding the Least Common Multiple by Continuous Division
- Warm Up! ............................................................................................................... 15
References ........................................................................................................................ 29
Least Common Multiple
One of the mathematical ideas you see in real-life is the
concept of fractions. Sometimes, you encounter fractions
where the numerator and the denominator have big
values. More often than not, we tend to manipulate these
numbers in such a way that they become small in number
but same in value. We use the concept of Greatest Common
Factor (GCF) to reduce these fractions in lowest terms.
Meanwhile, as we add or subtract dissimilar
fractions, we use the concept of Least Common
Multiple (LCM) so that the denominators will be
similar.
with simple to a much more complex
set of numbers and operations.
A foundation skill is essential as you
deal with far more advanced applications of mathematics. It is critical that we
learn the basics, such as the skill in finding the GCF and LCM of numbers, so
we can be ready to learn new things.
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splitting things into smaller sections,
equally distributing two or more sets of
items into their largest grouping, to
figure out how many people you can
invite into an occasion, arranging
something into rows or groups, or
figuring when something will happen again at the same time are real-life
situations or word problems that can be solved by finding the GCF and LCM.
You have encountered the skill in finding the GCF and LCM of two numbers
in 4th grade. In this unit, you shall learn the concepts and process in a more
in-depth manner, dealing with GCF or LCM of up to four numbers.
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STUDY GUIDE
Before you get started, answer the following items to help you assess your
prior knowledge and practice some skills that you will need in studying the
lessons in this unit.
a. 30 b. 42 c. 54 d. 72
2. Use factor tree to write the prime factorization of 24, 32, 58, and 64.
3. List the first 10 multiples of the following numbers:
a. 8 b. 12 c. 15 d. 20
At the end of this unit, you should be able to
find the common factors and greatest common factor of 2- to 3- digit
numbers using continuous division; and
find the common multiples and least common multiple of 2- to 3-digit
numbers using continuous division.
Finding the prime factorization of a number
Listing the first few multiples of a number
Identifying if a number is divisible by another number
Objectives
Instructions:
1. This activity may be done in groups.
2. Your teacher will hand over 12 blue popsicle sticks and 18 orange
popsicle sticks per group.
3. You are to divide these popsicle sticks equally into a number of
groups. What is the greatest number of groups in which the 12 blue
and 18 orange sticks are divided equally?
4. This time, your teacher will give you another 21 yellow popsicle sticks.
5. What is the greatest number of groups in which the 12 blue, 18
orange, and 21 yellow popsicle sticks are divided equally?
Lesson 1: Finding the Greatest Common Factor by
Continuous Division
Warm Up!
STUDY GUIDE
In the Warm Up! activity, you were able to group the 12 blue and 18 orange
sticks into 6, where there are 2 blue sticks and 3 orange sticks per group.
Moreover, when 21 yellow sticks were added, you changed the number of
groups into 3 where each group has 4 blue sticks, 6 orange sticks, and 7
yellow sticks.
The activity in Warm Up! dealt with the concept of greatest common factor
(GCF).
How do we find the GCF of two or more numbers?
Let us look at the following example.
Learn about It!
numbers. It is the highest number that
divides the given numbers without a
remainder.
factors that two or more numbers share.
Patrick needs to repack goods for donations
to charity. He has 24 kilos of rice, 12 packs of
noodles, and 36 cans of beans. He needs to
pack one kind in each box, with the same
amount of rice, noodles, and beans in each
box without excess. What is the greatest
number of packaged goods he can give to the charity?
To find the greatest number of packaged goods Patrick can give to the
charity, identify the greatest common factor of 24, 12, and 36.
One way to find the greatest common factor is to list the factors of each
number and to find the greatest among all the common factors.
The greatest common factor of 24, 12, and 32 is 12.
There is another way to find the GCF without listing down all the factors.
Continuous division is the process of continuously dividing a set of
numbers.
Find the GCF of 24, 12, and 36 through continuous division.
STUDY GUIDE
Step 1: Divide the given numbers by their common factor. A common
factor of 24, 12, and 36 is 2. Write this on the left side of the
given numbers. Write the quotients below the numbers.
Step 2: Repeat Step 1 until the remaining numbers do not have any
common factors except 1.
Step 3: Multiply the divisors (the numbers written vertically on the left).
GCF:
Therefore, the greatest number of packaged goods Patrick can
give to the charity is 12 with 2 kilograms of rice, 1 pack of
canned good, and 3 cans of beans per package.
STUDY GUIDE
Example 1: Find the greatest common factor of 20, 45, and 90.
Solution:
Step 1: Divide the given numbers by their common factor. Write this
on the left side of the given numbers. Write the quotients
below the numbers.
Step 2: Repeat step 1 until the remaining numbers do not have any
The numbers 4, 9, and 18 do not have any common factors
anymore, except 1.
GCF: 5
Thus, the greatest common factor of 20, 45, and 90 is 5.
Try It Yourself!
What is the greatest common factor of 11, 55, and 121?
Let’s Practice!
STUDY GUIDE
Example 2: Find the greatest common factor of 16, 20, and 32.
Solution:
below the numbers.
common factors.
GCF:
Thus, the greatest common factor of 16, 20, and 32 is 4.
Try It Yourself!
What is the greatest common factor of 12, 40, and 48?
Example 3: Find the greatest common factor of 8, 56, 84, and 112 using
continuous division.
Solution:
below the numbers.
STUDY GUIDE
GCF:
Therefore, the greatest common factor of 8, 56, 84, and 112 is
4.
Real-World Problems
different colors. She has 24 pink
bands, 36 yellow bands, 60 blue
bands, and 120 purple bands. She
wants to use it to make loom band
bracelets with the same number of loom bands for each
bracelet. If she will use all of her loom bands and all must have
the same colors and design, what is the greatest number of
loom band bracelets that Dorothy can make?
The greatest number of identical loom band bracelets that Dorothy
can make
24 pink bands
36 yellow bands
60 blue bands
120 purple bands
Step 3: Identify the concept to be used to solve the problem.
Greatest Common Factor (GCF)
GCF:
Therefore, Dorothy can make 12 identical loom band bracelets.
Each bracelet has 2 pink loom bands, 3 yellow loom bands, 5
blue loom bands, and 10 purple loom bands.
Roger is making sandwiches for his family’s picnic.
He has 48 ham slices and 72 cheese slices. What is
the greatest number of sandwiches he can make if
each sandwich has the same filling?
1. Find the GCF of the following numbers.
a. 12, 20, and 24
b. 16, 32, and 40
c. 45, 50, and 65
d. 10, 30, 50, and 75
e. 18, 22, 28, and 40
f. 50, 56, 72, and 90
2. Solve the given problems below.
a. Miss Fariscal has to prepare a number of seatworks for her lesson
for the day. She has four classes. One class has 40 students,
another class has 36, another class has 42, and another has 44.
Each class should have the same number of students working on
different problems. What must be the largest number of
seatworks she should make?
STUDY GUIDE
b. Mr. Maglalang, the Prefect of Students in a certain school, is
planning to have an educational tour for three grade levels with
750 students in Grade IV, 550 students in Grade V, and 400
students in Grade VI. What is the largest number of students per
group in each grade level so that each group has the same
number of students?
Instructions:
1. This is an activity for the whole class.
2. Everyone is given a paper with a printed happy face. Your teacher
assigns you a number from 1 until the last class number.
3. Together with the class, count starting from 1 and every time you
come to a multiple of 3, raise the paper with a happy face.
4. The difficulty of the activity will be increased. Your class will be split
into two. The first group raises their papers for multiples of 3, and
the second group raises their papers for multiples of 4.
5. In what numbers are two happy faces raised at the same time?
6. What is the least of these numbers?
Lesson 2: Finding the Least Common Multiple by
Continuous Division
Warm Up!
STUDY GUIDE
In the Warm Up! activity, there were two happy faces raised at the same time
when the numbers 12, 24, 36, 48, … are mentioned. These numbers are
multiples of both 3 and 4.
The least common multiple of 3 and 4 is 12.
Learn about It!
number that results from
whole number.
that are multiples of two or more
numbers. These are multiples that
two or more numbers share.
Definition 2.3: The least common multiple
(LCM) is the smallest number that
is divisible by the given numbers.
It is the smallest number that is a
multiple of the given numbers.
STUDY GUIDE
How do we find the LCM of two or more numbers? Let us take a look at the
problem below:
soccer every 7 days, and Robert plays
volleyball every 8 days. They were able to
play all today. How many days from now
will they be able to play all again?
Solution: Find the least common multiple of 6, 7, and 8 to determine the
number of days the three boys will be able to play again at the
same time.
Step 1: Divide the given numbers by their common factor. Write the
common factor on the left side of the given numbers. Write the
quotients below the numbers.
Note: If two numbers have a common factor and the other
numbers are not divisible by that common factor, divide the
two numbers by their common factor and copy the numbers
that are not divisible by that common factor.
Step 2: Repeat Step 1 until no two remaining numbers have any
common factor except 1.
STUDY GUIDE
Step 3: Multiply the divisors (the numbers written vertically on the left)
and the remaining numbers written horizontally in the last row.
LCM:
Thus, Riley, Allen, and Robert will be able to play all again after
168 days.
Example 1: Find the LCM of 10, 12, and 16.
Solution:
STUDY GUIDE
LCM:
Try It Yourself!
Example 2: Find the LCM of 60, 80 and 90.
Solution:
There is no need to include 1.
Therefore, the least common multiple (LCM) of 60, 80, and 90 is
720.
Example 3: Find the LCM of 27, 135, and 729.
Solution:
STUDY GUIDE
LCM:
Therefore, the least common multiple (LCM) of 27, 135, and
729 is 3 645.
Try It Yourself!
Find the LCM of 10, 25, 150, and 300 using continuous division.
buses going to Taguig City and
Makati City start to leave the bus
terminal. After the first buses
leave going to their destination,
buses going to Taguig leave
every 15 minutes and buses going to Makati leave every 20
minutes. How many minutes after 6:00 AM will the buses going
to Taguig and Makati leave the bus terminal at the same time
again?
Solution:
Step 1: Identify what is asked.
The number of minutes after 6:00 AM that the buses going to
Taguig and Makati will leave at the same time
Step 2: List down what are given.
6:00 AM
Taguig City: every 15 minutes
Makati City: every 20 minutes
Step 3: Identify the concept to be used to solve the problem.
Least Common Multiple (LCM)
LCM:
A bus going to Taguig and another bus going to Makati will
leave at the same time 60 minutes after 6:00 AM.
Try It Yourself!
The dance troupe conducts rehearsals every 11
days. The two groups are both having their
rehearsals in the gym today. How many days
from now will they have to share the gym again?
1. Determine the LCM of the following numbers.
a. 3, 15, 21
b. 25, 35, 100.
Check Your Understanding!
2. Solve the following problems.
a. What is the least number of candies that can be divided equally
among 8, 9, and 12 children?
b. You bring the drinks for your basketball team every sixth game.
Every third game is a home game. When will you first bring the
drinks to a home game? If there are 20 games in an annual
sportsfest, how many times will you bring the drinks to a home
game?
1. Give a set of three numbers with GCF of 15 and LCM of 90.
2. Identify two numbers that follow the requirements below:
a. sum is 13 and LCM is 36
b. GCF is 4 and LCM is 60
3. If the GCF of two numbers is 1, we call the two numbers as relatively
prime. Give a pair of relatively prime numbers whose LCM is 36.
4. Why do you think in finding the LCM using continuous division, it is
allowable to divide the numbers so long as at least two of them are
divisible by a given factor (and the others are copied), while in GCF, that
is not the practice?
STUDY GUIDE
You are a game developer who writes math questions. Your current game is
about GCF and LCM of numbers. Create a word problem finding the GCF and
LCM of numbers.
1. The word problems should involve the following:
a. different activities in your school. (e.g. schedule of recess time for
selected sections to avoid overcrowding in the canteen, alignment of
students in flag ceremony, etc.)
b. number word problem
2. Present your word problem with appropriate and creative illustrations
in the class and let your classmates answer it.
Performance Task Rubric
of the given numbers. Write the quotients
below the numbers.
have any common factor except 1.
Step 3: Multiply the divisors (the numbers written
vertically on the left).
factor.
any common factor except 1.
Step 3: Multiply the divisors (the numbers written
vertically on the left) and the remaining
numbers (the numbers written horizontally in
the last row).
Math Blaster. “GCF.” Accessed January 30, 2018.
http://www.mathblaster.com/parents/math-activities/gcf-activities
https://www.mathsisfun.com/least-common-multiple.html
References

the greatest common factor and the least common multiple

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