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Transcript
§ 4-3 Greatest Common Factor and Least
Common Multiple
Greatest Common Divisor
Given two whole numbers, how can we define the greatest common
divisor (GCD)?
Greatest Common Divisor
Given two whole numbers, how can we define the greatest common
divisor (GCD)?
Definition
The greatest common divisor of two non-zero whole numbers is the
largest whole number that divides both numbers.
Greatest Common Divisor
Given two whole numbers, how can we define the greatest common
divisor (GCD)?
Definition
The greatest common divisor of two non-zero whole numbers is the
largest whole number that divides both numbers.
Notation
gcd(a, b)
(a, b)
Finding Greatest Common Divisors
We will discuss 3 methods for finding greatest common divisors:
1
Set Intersections
Finding Greatest Common Divisors
We will discuss 3 methods for finding greatest common divisors:
1
Set Intersections
2
Prime Factorization
Finding Greatest Common Divisors
We will discuss 3 methods for finding greatest common divisors:
1
Set Intersections
2
Prime Factorization
3
the Euclidean Algorithm
The Set Intersection Method
Example
Find the gcd of 100 and 36.
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
Factors of 100
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
Factors of 100
{ 1, 2, 4, 5, 10, 20, 25, 50, 100 }
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
Factors of 100
{ 1, 2, 4, 5, 10, 20, 25, 50, 100 }
Factors of 36
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
Factors of 100
{ 1, 2, 4, 5, 10, 20, 25, 50, 100 }
Factors of 36
{ 1, 2, 3, 4, 6, 9, 12, 18, 36}
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
Factors of 100
{ 1, 2, 4, 5, 10, 20, 25, 50, 100 }
Factors of 36
{ 1, 2, 3, 4, 6, 9, 12, 18, 36}
The Set Intersection Method
Example
Find the gcd of 100 and 36.
To use this method, we list all of the factors each. This method is
good when neither whole number has a lot of factors and when the
numbers are such that we can quickly find all of the factors.
Factors of 100
{ 1, 2, 4, 5, 10, 20, 25, 50, 100 }
Factors of 36
{ 1, 2, 3, 4, 6, 9, 12, 18, 36}
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 =
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 =
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Model
We would use this model with whole numbers that we may not be
able to easily list all of the factors but when the prime factorization is
attainable.
Example
Find the gcd of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
gcd(2100, 6370) = 2 · 5 · 7 = 70
The Euclidean Algorithm
What did the division algorithm say?
The Euclidean Algorithm
What did the division algorithm say?
The Division Algorithm
For whole numbers a and b with a ≥ b, there exists unique whole
numbers q and r such that
a = bq + r
where 0 ≤ r < b
The Euclidean Algorithm
What did the division algorithm say?
The Division Algorithm
For whole numbers a and b with a ≥ b, there exists unique whole
numbers q and r such that
a = bq + r
where 0 ≤ r < b
We can use this to find greatest common divisors. Remember that
d|a, d|b ⇒ d|(a + b)
The Euclidean Algorithm
The Euclidean Algorithm
If a and b are whole numbers such that a = bq + r as described in the
Division Algorithm, then
gcd(a, b) = gcd(b, r)
The Euclidean Algorithm
The Euclidean Algorithm
If a and b are whole numbers such that a = bq + r as described in the
Division Algorithm, then
gcd(a, b) = gcd(b, r)
This is best used when we have large numbers that are not easy to find
the prime factorization for or when there is a large difference between
a and b.
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
4830 = 3786(1) + 1044
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
gcd(32766, 4830) = gcd(4830, 3786)
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
gcd(654, 390) = gcd(390, 264)
390 = 264(1) + 126
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
gcd(654, 390) = gcd(390, 264)
390 = 264(1) + 126
gcd(390, 264) = gcd(264, 126)
264 = 126(2) + 12
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
gcd(654, 390) = gcd(390, 264)
390 = 264(1) + 126
gcd(390, 264) = gcd(264, 126)
264 = 126(2) + 12
gcd(264, 126) = gcd(126, 12)
126 = 12(10) + 6
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
gcd(654, 390) = gcd(390, 264)
390 = 264(1) + 126
gcd(390, 264) = gcd(264, 126)
264 = 126(2) + 12
gcd(264, 126) = gcd(126, 12)
126 = 12(10) + 6
gcd(126, 12) = gcd(12, 6)
12 = 6(2) + 0
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
gcd(654, 390) = gcd(390, 264)
390 = 264(1) + 126
gcd(390, 264) = gcd(264, 126)
264 = 126(2) + 12
gcd(264, 126) = gcd(126, 12)
126 = 12(10) + 6
gcd(126, 12) = gcd(12, 6)
12 = 6(2) + 0
gcd(12, 6) = 6
The Euclidean Algorithm
Example
Find the greatest common divisor of 32766 and 4830.
32766 = 4830(6) + 3786
gcd(32766, 4830) = gcd(4830, 3786)
4830 = 3786(1) + 1044
gcd(4830, 3786) = gcd(3786, 1044)
3786 = 1044(3) + 654
gcd(3786, 1044) = gcd(1044, 654)
1044 = 654(1) + 390
gcd(1044, 654) = gcd(654, 390)
654 = 390(1) + 264
gcd(654, 390) = gcd(390, 264)
390 = 264(1) + 126
gcd(390, 264) = gcd(264, 126)
264 = 126(2) + 12
gcd(264, 126) = gcd(126, 12)
126 = 12(10) + 6
gcd(126, 12) = gcd(12, 6)
12 = 6(2) + 0
gcd(12, 6) = 6
So, gcd(32766, 4830) = 6.
A Theorem
Remember that if d|a and d|b then d|(a − b). We can use this when
finding greatest common divisors.
A Theorem
Remember that if d|a and d|b then d|(a − b). We can use this when
finding greatest common divisors.
Theorem
For whole numbers a and b, gcd(a, b) = gcd(a − b, b).
A Theorem
Remember that if d|a and d|b then d|(a − b). We can use this when
finding greatest common divisors.
Theorem
For whole numbers a and b, gcd(a, b) = gcd(a − b, b).
We can extend this as well.
Theorem
For whole numbers a and b and positive whole number k,
gcd(a, b) = gcd(a − kb, b),
A Theorem
Remember that if d|a and d|b then d|(a − b). We can use this when
finding greatest common divisors.
Theorem
For whole numbers a and b, gcd(a, b) = gcd(a − b, b).
We can extend this as well.
Theorem
For whole numbers a and b and positive whole number k,
gcd(a, b) = gcd(a − kb, b),
This theorem is the reason that the Euclidean Algorithm really works.
Least Common Multiples
Who can define the least common multiple of two whole numbers a
and b?
Least Common Multiples
Who can define the least common multiple of two whole numbers a
and b?
Least Common Multiple (LCM)
For whole numbers a and b, the least common multiple of a and b is
te smallest whole number with a and b as factors.
Least Common Multiples
Who can define the least common multiple of two whole numbers a
and b?
Least Common Multiple (LCM)
For whole numbers a and b, the least common multiple of a and b is
te smallest whole number with a and b as factors.
Notation:
1
lcm(a, b)
Least Common Multiples
Who can define the least common multiple of two whole numbers a
and b?
Least Common Multiple (LCM)
For whole numbers a and b, the least common multiple of a and b is
te smallest whole number with a and b as factors.
Notation:
1
lcm(a, b)
2
[a, b]
Number Line Model
At first introduction, we could use number lines to explicitly illustrate
to students how to find lcms.
Number Line Model
At first introduction, we could use number lines to explicitly illustrate
to students how to find lcms.
Example
Find the least common multiple of 2 and 5.
0
1
2
3
4
5
6
7
8
9
10
Number Line Model
At first introduction, we could use number lines to explicitly illustrate
to students how to find lcms.
Example
Find the least common multiple of 2 and 5.
0
1
2
3
4
5
6
7
8
9
10
Number Line Model
At first introduction, we could use number lines to explicitly illustrate
to students how to find lcms.
Example
Find the least common multiple of 2 and 5.
0
1
2
3
4
5
6
7
8
9
10
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 =
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 =
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
Prime Factorization Method
The number line method would be very tedious, however, with larger
numbers, and we wouldn’t want to use an illustrated method with
older students. But we can always go back to prime factorizations.
Example
Find the LCM of 2100 and 6370.
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
lcm(2100, 6370) = 22 · 3 · 52 · 72 · 13
LCM-GCD Method
LCM-GCD Method
For whole numbers a and b,
a · b = lcm(a, b) · gcd(a, b)
Why?
2100 =
LCM-GCD Method
LCM-GCD Method
For whole numbers a and b,
a · b = lcm(a, b) · gcd(a, b)
Why?
2100 = 22 · 3· 52 · 7
LCM-GCD Method
LCM-GCD Method
For whole numbers a and b,
a · b = lcm(a, b) · gcd(a, b)
Why?
2100 = 22 · 3· 52 · 7
6370 =
LCM-GCD Method
LCM-GCD Method
For whole numbers a and b,
a · b = lcm(a, b) · gcd(a, b)
Why?
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
LCM-GCD Method
LCM-GCD Method
For whole numbers a and b,
a · b = lcm(a, b) · gcd(a, b)
Why?
2100 = 22 · 3· 52 · 7
6370 = 2· 5· 72 · 13
2100 · 6370 = 23 · 3 · 53 · 73 · 13
LCM-GCD Method
So how do we use it? It is sometimes easier to find the factors than the
multiples ...
Example
Find the LCM of 3420 and 4290.
LCM-GCD Method
So how do we use it? It is sometimes easier to find the factors than the
multiples ...
Example
Find the LCM of 3420 and 4290.
We again begin with prime factorization.
LCM-GCD Method
So how do we use it? It is sometimes easier to find the factors than the
multiples ...
Example
Find the LCM of 3420 and 4290.
We again begin with prime factorization.
3420 = 22 · 32 · 5 · 19
LCM-GCD Method
So how do we use it? It is sometimes easier to find the factors than the
multiples ...
Example
Find the LCM of 3420 and 4290.
We again begin with prime factorization.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
LCM-GCD Method
So how do we use it? It is sometimes easier to find the factors than the
multiples ...
Example
Find the LCM of 3420 and 4290.
We again begin with prime factorization.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
Then, the product: 3420 · 4290 = 14671800
LCM-GCD Method
So how do we use it? It is sometimes easier to find the factors than the
multiples ...
Example
Find the LCM of 3420 and 4290.
We again begin with prime factorization.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
Then, the product: 3420 · 4290 = 14671800
Note: This is a multiple of our two numbers, just not necessarily the
smallest one. When would the product be the LCM?
LCM-GCD Method
Now that we have the product, we need to find the GCD.
LCM-GCD Method
Now that we have the product, we need to find the GCD.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
LCM-GCD Method
Now that we have the product, we need to find the GCD.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
gcd(3420, 4290) = 2 · 3 · 5 = 30
LCM-GCD Method
Now that we have the product, we need to find the GCD.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
gcd(3420, 4290) = 2 · 3 · 5 = 30
So, we can now find the LCM.
14671800 = 30 · lcm(3420, 4290)
LCM-GCD Method
Now that we have the product, we need to find the GCD.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
gcd(3420, 4290) = 2 · 3 · 5 = 30
So, we can now find the LCM.
14671800 = 30 · lcm(3420, 4290)
14671800
lcm(3420, 4290) =
30
LCM-GCD Method
Now that we have the product, we need to find the GCD.
3420 = 22 · 32 · 5 · 19
4290 = 2 · 3 · 5 · 11 · 13
gcd(3420, 4290) = 2 · 3 · 5 = 30
So, we can now find the LCM.
14671800 = 30 · lcm(3420, 4290)
14671800
lcm(3420, 4290) =
30
lcm(3420, 4290) = 489060
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
12
18
30
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
2
12
18
30
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
2
12
18
30
6
9
15
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
2
12
18
30
3
6
9
15
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
2
12
18
30
3
6
9
15
2
3
5
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
2
12
18
30
3
6
9
15
2
3
5
LCM of More Than 2 Whole Numbers
We do not have to find the LCM of 3 or more numbers separately and
then combine. There are ways we can do it in one shot. One way is
using a common factor grid.
Example
Find the LCM of 12, 18 and 30.
2
12
18
30
3
6
9
15
2
3
5
lcm(12, 18, 30) = 22 · 32 · 5 = 180
