Download Activity #36

Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #36 – GCF & LCM
California State Content Standard – Number Sense – Grade Six
2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division:
2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems
with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).
Definition- Greatest Common Factor
Let m and n be natural numbers. The greatest natural number d that divides both m and n is called their greatest
common factor and we write d = GCF(m , n).
** FINDING THE GCF USING THE SET INTERSECTION METHOD **
 Example:
Find the GCF of 24 and 30, sometimes written GCF(24 , 30), using the set intersection method.
Solution:
M24 : { 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24 }
M30 : { 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30 }
M24  M30 : { 1 , 2 , 3 , 6 }
So, the greatest common factor in the intersection set is 6. Thus, GCF(24 , 30) = 6.
Find the GCF of each of the following sets of numbers using the set intersection method.
1)
GCF(54 , 72)
2)
GCF(27 , 36)
3)
GCF(160 , 188)
4)
GCF(28 , 42 , 98)
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Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #36 – GCF & LCM
** FINDING THE GCF USING PRODUCT OF PRIMES REPRESENTATION **
 Example:
Find the GCF of 24 and 30 using product of primes representation.
3
2 6
Solution: Use Prime Factorization
24:
30:
2 12
5
3 15
So, 24 = 2223 or can be written 233
2 30
2 24
So, 30 = 235
3
So, we have 24 = 2 3 and 30 = 235.
Choosing the smaller of the exponents on each prime, we see that GCF(24 , 30) = 2131 = 6
Note that 5 was not included, because it was not represented in the factors of 24. So, technically we can write
24 = 23350. So, when we choose the smaller of the exponents we would choose 50, which is simply 1.
 Exercises:
Find the GCF of each of the following sets of numbers using product of primes representation.
5)
GCF(120 , 144)
6)
GCF(156 , 182)
** FINDING THE GCF USING A VENN DIAGRAM **
 Example:
Find the GCF of 24 and 30, using a Venn diagram.
Solution: Use product of primes representation.
3
2 6
2
2 12
5
3 15
24
2 30
So, we have 24 = 2223 and 30 = 235.
Thus, the GCF is the product of all prime factors within
the intersection of the two sets of the Venn diagram.
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Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #36 – GCF & LCM
 Exercises:
Find the prime factorization of each number. Then place the prime factors in the approximate part of the
Venn diagram and find the GCF of the numbers.
7)
GCF (54 , 72) = ____________________
8)
GCF (84 , 56) = ____________________
9)
GCF (21 , 10) = ____________________
10)
GCF (18 , 45 , 60) = ___________________
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Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #36 – GCF & LCM
Definition- Least Common Multiple
Let a and b be natural numbers. The least natural number m that is a multiple of both a and b is called their
least common multiple and we write m = LCM(a , b).
** FINDING THE LCM USING THE SET INTERSECTION METHOD **
 Example:
Find the LCM of 24 and 30, sometimes written LCM(24 , 30), using the set intersection method.
Solution:
M24 : { 24 , 48 , 72 , 96 , 120 , 144 , 168 , 192 , 216 , 240 , 264 , 288 , …………}
M30 : { 30 , 60 , 90 , 120 , 150 , 180 , 210 , 240 , 270 , 300 , …………..}
M24  M30 : { 120 , 240 , ………}
So, the least common number in the intersection set is 120. Thus, LCM(24 , 30) = 120.
 Exercises:
Find the LCM of each of the following sets of numbers using the set intersection method.
11)
LCM(54 , 72)
12)
LCM(27 , 36)
13)
LCM(16 , 18)
14)
LCM(8 , 9 , 12)
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Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #36 – GCF & LCM
** FINDING THE LCM USING PRODUCT OF PRIMES REPRESENTATION **
 Example:
Find the LCM of 24 and 30 using product of primes representation.
3
2 6
Solution: Use Prime Factorization
24:
30:
2 12
5
3 15
So, 24 = 2223 or can be written 233
2 30
2 24
So, 30 = 235
3
So, we have 24 = 2 3 and 30 = 235.
Choosing the larger of the exponents on each prime, we see that LCM(24 , 30) = 233151 = 120
 Exercises:
Find the LCM of each of the following sets of numbers using product of primes representation.
15)
LCM(42 , 48)
16)
LCM(51 , 68)
17)
LCM(48 , 160)
18)
LCM(6 , 15 , 21)
** FINDING THE LCM USING A VENN DIAGRAM **
 Example:
Find the LCM of 24 and 30, using a Venn diagram.
Solution: Use product of primes representation.
3
2 6
2
2 12
5
3 15
24
2 30
So, we have 24 = 2223 and 30 = 235.
Thus, the LCM is the product of all prime factors within
the union of the two sets of the Venn diagram.
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Math 104 - Cooley
Math For Elementary Teachers I
OCC
Activity #36 – GCF & LCM
 Exercises:
Find the prime factorization of each number. Then place the prime factors in the approximate part of the
Venn diagram and find the LCM of the numbers.
19)
LCM (54 , 72) = ____________________
20)
LCM (84 , 56) = ____________________
21)
LCM (9 , 16) = ____________________
22)
LCM (24 , 90 , 100) = __________________
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