Download Unit 3 - LCM

Least Common Multiple (LCM)
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