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GREATEST COMMON
FACTORS (GCF) & LEAST
COMMON MULTIPLES (LCM)
DEFINITION
•
Greatest Common Factor
(GCF) – The largest number
that divides exactly into two or
more numbers
FINDING THE GCF
Begin with two numbers.
30 and 84
Step 1 – Complete the prime factorization
of these numbers. Do not use exponents.
30
84
2 x 3 x 5 =30
2 x 2 x 3 x 7=84
FINDING THE GCF
Step 2 – Write the prime factorizations underneath each other
2 x 3 x 5=30
2 x 2 x 3 x 7=84
Step 3 – Circle each number that appears in both factorizations
2 x 3 x 5 =30
2 x 2 x 3 x 7=84
FINDING THE GCF
Note-The second 2 is not circled in the bottom prime factorization
because 2 only appears once in the first factorization.
2 x 3 x 5 =30
2 x 2 x 3 x 7=84
Step 4-Multiply the circled numbers.
2 x 3=6
6 IS YOUR GCF FOR 30 AND 84!
PRACTICE
Find the GCF of 24 and 96
Step 1-Prime Factorization.
24
96
2 x 2 x 2 x 3=24
2 x 2 x 2 x 2 x 2 x 3=96
Step 2-Write the prime factorizations underneath each other.
2 x 2 x 2 x 3=24
2 x 2 x 2 x 2 x 2 x 3=96
PRACTICE
Find the GCF of 24 and 96
Step 3-Circle the common numbers.
2 x 2 x 2 x 3=24
2 x 2 x 2 x 2 x 2 x 3=96
Step 4-Multiply the circled numbers.
2 x 2 x 2 x 3=24
24 IS YOUR GCF FOR 24 AND 96!
PRACTICE
Find the GCF of 18 and 35
Step 1-Prime Factorization.
18
35
2 x 3 x 3=18
5 x 7=35
Step 2-Write the prime factorizations underneath each other.
2 x 3 x 3=18
5 x 7=35
PRACTICE
Find the GCF of 18 and 35
Step 3-Circle the common numbers.
2 x 3 x 3=18
5 x 7=35
WE DON’T HAVE ANY COMMON NUMBERS!
Note-If the prime factorizations have no numbers in common,
the GCF is 1.
1 IS YOUR GCF FOR 18 AND 35!
PRACTICE
Find the GCF of 15 and 36
Step 1-Prime Factorization.
15
36
3 x 5=15
2 x 2 x 3 x 3=36
Step 2-Write the prime factorizations underneath each other.
3 x 5=15
2 x 2 x 3 x 3=36
PRACTICE
Find the GCF of 15 and 36
Step 3-Circle the common numbers.
3 x 5 =15
2 x 2 x 3 x 3=36
Note-If the prime factorizations only have one number in
common, that number is your GCF.
3 IS YOUR GCF FOR 15 AND 36!
FINDING THE GCF FOR MORE THAN 2 NUMBERS
•
To find the GCF for more than 2
numbers, take the same steps as
before, but you will circle the
common numbers in all of the
prime factorizations.
FINDING THE GCF FOR MORE THAN 2 NUMBERS
Begin with the given numbers.
24, 48, and 60
Step 1 – Complete the prime factorization
of these numbers. Do not use exponents.
24
48
60
2 x 2 x 2 x 3 =24 2 x 2 x 2 x 2 x 3=48 2 x 2 x 3 x 5=60
FINDING THE GCF FOR MORE THAN 2 NUMBERS
Step 2 – Write the prime factorizations underneath each other
2 x 2 x 2 x 3 = 24
2 x 2 x 2 x 2 x 3 = 48
2 x 2 x 3 x 5 = 60
Step 3 – Circle each number that appears in all factorizations
2 x 2 x 2 x 3 = 24
2 x 2 x 2 x 2 x 3 = 48
2 x 2 x 3 x 5 = 60
FINDING THE GCF FOR MORE THAN 2 NUMBERS
Step 4-Multiply the circled numbers.
2 x 2 x 3=12
12 IS YOUR GCF FOR 24, 48, AND 60!
Note-If no number appears in all prime factorizations, then
GCF is 1.
SPECIAL EXAMPLE
Find the GCF of 12, 16, 31, 44, 68, and 96.
•
To solve this problem, you could do all of the
prime factorizations for the numbers listed.
However, 31 is a prime number. Whenever a
prime number is in your given set of numbers,
and all the other numbers are not multiples of
the prime number, then the GCF is always 1.
NOTES
•
•
•
•
•
Carefully check that you have circled all of the
matching prime factors and haven’t circled any nonmatching ones.
If the set of given numbers does not have any prime
factors in common, the GCF is 1.
If the set of given numbers only has one prime factor
in common, that prime factor is your GCF.
To find the GCF for more than two numbers, the prime
factors must be present in all prime factorizations.
If one (or more) of the numbers in the given set is a
prime number, and all of the other numbers are not
multiples of the prime number, the GCF is always 1.
Questions?
DEFINITION
•
Least Common Multiple (LCM)
– The smallest number that is
a multiple of two or more
numbers.
FINDING THE LCM-BASIC METHOD
Begin with two numbers.
12 and 80
Step 1 – List several multiples of each number.
12
12, 24, 36, 48, 60, 72, 84, 96, 108, 120,
132, 144, 156, 168, 180, 192, 204, 216,
228, 240, 252, 264, 276, 288, 300
80
80, 160, 240, 320, 400
FINDING THE LCM-BASIC METHOD
Step 2-Find the smallest number that occurs in both lists.
12
12, 24, 36, 48, 60, 72, 84, 96, 108, 120,
132, 144, 156, 168, 180, 192, 204, 216,
228, 240, 252, 264, 276, 288, 300
80
80, 160, 240, 320, 400
240 is you LCM for 12 and 80!
Note-You will always have to list more multiples of the smaller number.
PRACTICE
Find the LCM of 16 and 52
16
16, 32, 48, 64, 80, 96, 112, 128, 144,
160, 176, 192, 208, 224, 240, 256, 272
52
52, 104, 156, 208, 260, 312
208 is your LCM for 16 and 52
PRACTICE
Find the LCM of 17 and 19
17
17, 34, 51, 68, 85, 102, 119, 136, 153,
170, 187, 204, 221, 238, 255, 272, 289
306, 323, 340, 357, 374, 391, 408, 425
19
19, 38, 57, 76, 95, 114, 133, 152, 171,
190, 209, 228, 247, 266, 285, 304, 323,
342, 361, 380, 399, 418, 437, 456
323 is your LCM for 17 and 19
Note-The LCM of two prime numbers is always the product of the numbers.
FINDING THE LCM FOR MORE
THAN 2 NUMBERS-BASIC METHOD
•
To find the LCM for more than 2
numbers, take the same steps as
before, but you will circle the first
multiple that is present in all
lists.
FINDING THE LCM FOR MORE
THAN 2 NUMBERS-BASIC METHOD
Find the LCM of 8, 30, and 48
8
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88,
96, 104, 112, 120, 128, 136, 144, 152
160, 168, 176, 184, 192, 200, 208, 216,
224, 232, 240, 248, 256, 264, 272, 280
30
30, 60, 90, 120, 150, 180,
210, 240, 270, 300, 330
48
48, 96, 144, 192, 240,
288, 336, 384, 432, 480
240 is your LCM for 8, 30, and 48
FINDING THE LCM-ADVANCED METHOD
Begin with two numbers.
12 and 80
Step 1 – Complete the prime factorization
of these numbers. Do not use exponents.
12
2 x 2 x 3 =12
80
2 x 2 x 2 x 2 x 5=80
FINDING THE LCM-ADVANCED METHOD
Step 2 – Write the prime factorizations underneath each other
2 x 2 x 3=12
2 x 2 x 2 x 2 x 5= 80
Step 3 – Count the times each factor appears in each factorization.
2 appears twice, 3 appears once
2 x 2 x 3=12
2 appears four times, 5
2 x 2 x 2 x 2 x 5= 80
appears once
FINDING THE LCM-ADVANCED METHOD
Step 4-Use the highest occurrence of each number to create
a multiplication problem.
2 appears twice, 3 appears once
2 x 2 x 3=12
2 appears four times, 5
2 x 2 x 2 x 2 x 5= 80
appears once
2 x 2 x 2 x 2 x 3 x 5=
2 appears four
times in the second 3 appears once
in the first
factorization
factorization
5 appears once
in the second
factorization
FINDING THE LCM-ADVANCED
Step 5 – Solve the problem
2 x 2 x 2 x 2 x 3 x 5= 240
240 is you LCM for 12 and 80!
PRACTICE
Find the LCM of 12 and 26
Step 1 – Prime factorization
12
26
2 x 2 x 3 =12
2 x 13 =26
Step 1 – Write the prime factorizations underneath each other.
2 x 2 x 3 =12
2 x 13 =26
PRACTICE
Find the LCM of 12 and 26
Step 3 – Count the times each factor appears in each factorization.
2 x 2 x 3 =12
2 x 13 =26
2 appears twice, 3 appears once
2 appears once, 13 appears once
Step 4-Use the highest occurrence of each number to create
a multiplication problem.
2 x 2 x 3 x 13=
2 appears twice in the
3 appears once in the
first factorization
first factorization
13 appears once in the
second factorization
FINDING THE LCM-ADVANCED
Step 5 – Solve the problem
2 x 2 x 3 x 13=156
156 is you LCM for 12 and 26!
NOTES
The LCM of two prime numbers is always the
product of the two numbers.
 The basic method is better for smaller
numbers.
 The advanced method is better for larger
numbers.

Questions?