Download Harford Community College – MATH 017 Worksheet: Finding the

Harford Community College – MATH 017
Worksheet: Finding the LCM of a Group of Numbers
Finding the LCM (least common multiple) of a set of numbers is a tool that you will need
for this class and probably for other math classes. One can also find the LCM when
variables are involved, but this will be covered later.
LCM: Given a group of two or more numbers, the lowest number that the group shares
as a common factor. In other words, the lowest number that all members of the group
divide into evenly.
While it is true that multiplying the group of numbers together will yield a common
multiple, it may not be the LEAST common multiple.
Example 1: Find the LCM of 6 and 8.
Method 1: Start with the largest number in the group (in this case 8) and enumerate its
multiples until you reach a number that all members of the group divide into evenly.
We begin with the first multiple of 8 which is 8  1  8 . Does 6 evenly divide into 8?
No.
The next multiple of 8 is 8  2  16 . Does 6 evenly divide into 16? No.
The next multiple of 8 is 8  3  24 . Does 6 evenly divide into 24? Yes.
Therefore, the LCM of 6 and 8 is 24 because:
8, 16, 24
6, 12, 18, 24
Method 2: Use a tree to break each number into its prime factorization.
Recall that a prime number is a natural number that is divisible only by itself and 1;
otherwise, the number is composite. By convention, 1 is not considered a prime or
composite number. Thus, if we have the set-builder notation: {x | x is a prime number},
we get 2,3,5,7,11,13,17,19,23,29,31, in roster notation. You should memorize the first
ten prime numbers.
To find the prime factorization of a number, start with that number at the top of the tree.
If the number is prime, stop – you are done; otherwise, draw two branches downward.
On the left branch, find a prime that will evenly divide the number and circle it. On the
right branch, write down how many times the prime evenly divides the number. Nothing
needs to be done with the number on the left branch since it is prime. If the number on
the right branch is also prime, circle it – you are done; otherwise, repeat the process until
both branches hold prime numbers. The prime factorization then is the product of the
circled numbers. It seems complicated in words, but after drawing a few prime
factorization trees, the process should become clearer.
Going back to our example, let us draw the prime factorization tree for 6. Is 6 prime?
No. Then draw two branches extending from 6 and find a prime factor. Note that it does
not matter which prime factor you choose – a number can have a variety of prime
factorization trees, but only one prime factorization. You will get the original number if
you multiply the circled prime numbers together in any order. This is an example of the
commutative property of multiplication at work.
6
/
\
2
3
2 is prime since it is on the left branch. Now look at the number on the right branch. Is 3
prime? Yes, the numbers on both branches are prime so you are done. Thus, the final
prime factorization tree is (I use red to indicate circled numbers):
6
/
\
2
3
And the prime factorization is 2  3 since 2  3 = 6. As indicated by the definition of
prime factorization, all numbers in a prime factorization must be prime!
Let us do the prime factorization tree for 8. First, 8 is not prime so we extend two
branches off of it. The only prime number that evenly divides 8 is 2. Thus:
8
/
\
2
4
Is 4 a prime? No, so extend two branches from 4 and find a prime number that evenly
divides 4. Again, this number is 2:
8
/
\
2
4
/
2
\
2
Both numbers are prime so we stop and the prime factorization of 8 is 2  2  2 since
2  2  2 = 8. While this is correct, to shorten the amount of numbers written, we often
use exponential notation. Thus, the prime factorization of 8 written in exponential
notation is 23 .
Once we have the prime factorizations of all numbers in the group, we rewrite the prime
factorizations such that each prime factor shares its own column. Next, circle the largest
number in each column. Finally, multiply the circled numbers to arrive at the LCM.
6:
2
8:
23
3
23  3  24 which agrees with the result obtained by using the first method. You may be
asking yourself “why should I use the second method when the first method is so much
faster?” First, the prime factorization method only seemed complex since I explained
every step. When you get used to the method, you will find that it is equally as fast as
the first method. Secondly, the first method was simple in the first example since the
numbers were small. However, it quickly becomes cumbersome when the numbers
become larger. Further, the first method becomes more complex when we try to find the
LCM of more than two numbers. For these reasons, I personally prefer the method of
prime factorization. However, it is up to you to decide which method better suits you –
you should be able to consistently find the correct LCM with whichever method you
choose.
Some divisibility rules that may be helpful:
A number is divisible by 2 when its ones digit ends in either 0, 2, 4, 6, or 8.
Examples: 10, 22, 34, 46, 58.
A number is divisible by 3 when the sum of its digits is divisible by 3.
Examples: 18, 405, 1113.
A number is divisible by 5 when its ones digit ends in either 0 or 5.
Examples: 50, 305.
Now, let us work an example when the numbers are large.
Example 2: Find the LCM of 100 and 104.
Method 1:
104 x 1 = 104
104 x 2 = 208
104 x 3 = 312
104 x 4 = 416
104 x 5 = 520
:
Does 100 evenly divide 104?
Does 100 evenly divide 208?
Does 100 evenly divide 312?
Does 100 evenly divide 416?
Does 100 evenly divide 520?
No.
No.
No.
No.
No.
104 x2600 = 270,400
Does 100 evenly divide 270,400? Yes, finally!
Method 2:
104
/
\
2
52
/
\
2
26
/
\
2
13
The prime factorization of 104 is then 23  13 .
100
/
\
2
50
/
\
2
25
/
5
\
5
The prime factorization of 100 is then 2 2  52 .
Next, rewrite the prime factors so that each shares its own column. Then circle the
largest number in each column:
104:
23
100:
22
13
52
Multiply the circled numbers to get the LCM of 100 and 104 which is 2600. You be the
judge – which method required less work in this previous example?
Now let us look at an example when we try to find the LCM of 3 numbers. Remember
that in this case that all 3 numbers must divide evenly into the LCM.
Example 3: Find the LCM of 2, 14, and 25.
Method 1:
25 x 1 = 25
25 x 2 = 50
25 x 3 = 75
Do both 2 and 14 evenly divide 25? No.
Do both 2 and 14 evenly divide 50? 2 does, but 14 does not - no.
Do both 2 and 14 evenly divide 75? No.
25 x 4 = 100
25 x 5 = 125
:
25 x 350 = 8750
Do both 2 and 14 evenly divide 100? 2 does, but 14 does not - no.
Do both 2 and 14 evenly divide 75? No.
Do both 2 and 14 evenly divide 8750? Yes, finally!
Method 2:
2 is already prime so there is no need to do a prime factorization tree for it. The prime
factorization of 2 is 2.
14
/
\
2
7
The prime factorization of 14 is 2  7 .
25
/
5
\
5
The prime factorization of 25 is 52.
Next, rewrite the prime factors so that each shares its own column. Then circle the
largest number in each column:
52
25:
14:
2
2 :
2
7
Therefore, the LCM of 2, 14, and 25 is 350. If there is a column where the largest
number occurs multiple times, select only one of the numbers – it does not matter which
one you select since they are both the same. Note that this situation occurs in 2s column
of the last example.
What follows are a set of exercises to test your understanding of the LCM concept. The
answers follow on the last page.
Exercise Set:
Directions: Find the LCM of each group of numbers.
1) 3 and 15
2) 6 and 7
3) 4 and 6
4) 12 and 14
5) 20 and 25
6) 18 and 30
7) 4, 5, and 6
8) 7, 8, and 9
9) 10, 12, and 14
10) 15, 20, and 30
11) 21, 35, and 45
12) 60, 80, and 100
13) 2, 4, 6, and 8
14) 3, 5, 7, and 9
15) 18, 22, 26, 32, and 40
Solutions are on the next page
Solutions:
1) 15
2) 42
3) 12
4) 84
5) 100
6) 90
7) 60
8) 504
9) 420
10) 60
11) 315
12) 1200
13) 24
14) 315
15) 205,920