Download Unit 3 - LCM and GCF

Least Common Multiple (LCM) & Greatest Common Factor (GCF)
The Least Common Multiple or two or more numbers, a.k.a. the LCM, is the
smallest multiple that is common to the two or more numbers.
Example: Find the LCM of 12 and 20 by listing their multiples.
Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, etc
Multiples of 20 are 20, 40, 60, 80, 100, 120, 140, etc.
From these two lists, we see common multiples of 60 and 120 and if we kept
listing, we would see more common multiples. But since 60 is the smallest
common multiple, 60 is the Least Common Multiple. i.e. LCM = 60.
Why must I learn how to find the LCM? The LCM is what makes up the least
common denominator needed to add fractions. For example, if we add the
fractions below, we need a common denominator. The least common
denominator will be 20, the LCM of 5 and 4.
Method 1 for finding LCM of a list of numbers: List
multiples of the largest number of the list and use the
first multiple divisible by all the numbers.
Example: Find the LCM of 15, 10, and 40.
Multiples of 40 are 40, 80, 120 . . . Since 120 is divisible by both 15 and 10, we
can stop listing and conclude that the LCM is 120.
Example: Find the LCM of 15 and 19.
Multiples of 19 are 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247,
266, 285, . . .
Finally, when we get to the multiple of 285, we find the LCM!
We see here, that Method 1 does not work all that well! So we have a second
method.
Method 2 for finding LCM of a list of numbers: Write out
the prime factorizations. Then use the product of all the
largest powers of factors in these factorizations.
Example: Find the LCM of 90, 40, and 17.
To find the prime factorizations, you rewrite the number as a product of numbers
such that all of the numbers are prime numbers. Prime numbers are not divisible
by any other whole number but themselves.
90 = 2  45 = 2  3  15 = 2  3  3  5
The prime factorization of 30 is 21  32  51
40 = 2  20 = 2  2  10 = 2  2  2  5
The prime factorization of 40 is 23  51
17 = 1  17
The prime factorization of 17 is 1  171
The largest power of 2 is 23. The largest power of 3 is 32. The largest power of 5
is 51. The largest power of 17 is just 171. The product of these powers is 23  32 
51  171 = 8  9  5  17= 6120.
We conclude that the LCM of 90, 40, and 17 is 6120.
Why must we learn Method 2? It seems complicated!
Answer: Algebraic Fractions. To add the
fractions shown here we must factor the
denominators and then use the product of
the largest power or each factor present as our least common denominator. We
will not add such fractions in this class, but you will if you take Algebra II.
CONTINUED ON NEXT PAGE . . .
Example: Find the LCM of 44 and 80.
44 = 2  22 = 2  2  11 so the prime factorization is 22 11
80 = 2  40 = 2  2  20 = 2  2  2  10 = 2  2  2  2  5
so the prime factorization is 245
The LCM will be 24  5  11 = 16  5  11 = 880
Example: Find the LCM of 17 and 13.
17 = 17  1 and 13 = 13  1 so the LCM will be 13  17 = 221
NOTE: 17 and 13 are prime numbers since no other number but 1 divides into
them.
Example: Find the LCM of 13, 11 and 3.
13 = 13  1, 11 = 11  1 and 3 = 3  1
so the LCM will be 13  11  3 = 429
In general, if all of the numbers are prime, then the LCM will be the product
of all the numbers given.
Example: The LCM of 11, 17, and 2 is 11  17  2 = 374
Greatest Common Factor (GCF)
The Greatest Common Factor, or GCF for short, of a list of numbers is the
largest factor that divides into all of the numbers evenly.
Example: The GCF of 20 and 30 is 10 since 10 divides into 20 evenly 2 times
and 10 divides into 30 evenly 3 times.
Where do we use the GCF? The GCF is used when we reduce a fraction to
lowest terms. For example, we reduce the fraction 20/30 by dividing out the GCF
of 10 from both numerator and denominator, as shown below. The result is a
reduced fraction of 2/3.
The quickest way to reduce a fraction is to divide out by the GCF!
Method 1 for finding GCF of a list of numbers: List all
the possible factors of the numbers. Choose the largest
common factor from the lists.
Example: Find the GCF of 24 and 40
Factors of 24 are all the numbers that divide in evenly. We can find these by
listing out all the multiplied products that make 24.
24 = 1 x 24
24 = 2 x 12
24 = 3 x 8
24 = 4 x 6
24 = 6 x 4
24 = 8 x 3
24 = 12 x 2 Notice that we could have stopped at 4x6 since we just repeated
what we had in reverse order. The factors are {1,2,3,4,6,8,12,24}
In a similar manner 40 = 1 x 40 = 2 x 20 = 4 x 10 = 5 x 8
The factors of 40 are {1, 2,4,5,8,10, 20}
The largest number common to both lists is 8, so the GCF of 24 and 40 is 8.
Application: The fraction 24/40 reduces to 3/5 when we divide 24 by 8 & 40 by 8.
Example: Find the GCF of 13 and 20
The factors of 13 are {1, 13} since 13. The factors of 20 are {1,2,4,5,10, 20}
The largest number common to both lists is 1 so the GCF is 1.
Applied to fractions, 13/20 would not be able to be reduced !
Method 2 for finding GCF of a list of numbers: Write the
prime factorizations of the numbers. The GCF will be
product of common factors present in all the prime
factorizations.
Example: Find the GCF of 44 and 80.
44 = 2  22 = 2  2  11
80 = 2  40 = 2  2  20 = 2  2  2  10 = 2  2  2  2  5
There are two factors of 2 in both lists, so the GCF is 2  2 = 4
Example: Find the GCF of 24 and 48
24 = 2  12 = 2  2  6 = 2  2  2  3
48 = 2  24 = 2  2  12 = 2  2  2  6 = 2  2  2  2  3
There are three factors of 2 and one factor of 3 in both lists.
The GCF will be 2  2  2  3 = 24
Application of this: 24/48 reduces to ½ since we can divide the GCF of 24 out of
both the numerator and denominator.
Why Method 2 For Finding the GCF?
Many students find this second method for finding the GCF a bit confusing. But
we absolutely need this method for reducing an algebraic fraction like the one
below that you will see in an Algebra II class. In the problem below, we would
reduce the fraction by writing the top and bottom in factored form and then start
canceling out factors common to both top and bottom
In Perspective:
If you are not going to take any more algebra classes ever, then Method 1 for
both LCM and GCF will probably be sufficient for what you need. However, if you
plan to take more algebra, particularly Algebra II, you should learn Method 2.