Transcript least common multiple (LCM)

5-1 Least Common Multiple
Learn to find the least common multiple
(LCM) of a group of numbers.
5-1 Least Common Multiple
Vocabulary
least common multiple (LCM)
5-1 Least Common Multiple
Remember!
A multiple of a number is the
product of the number and any
nonzero whole number.
5-1 Least Common Multiple
Additional Example 1: Consumer Application
English muffins come in packs of 8, and eggs come in
cartons of 12. If there are 24 students, what is the least
number of packs and cartons needed so that each
student has a muffin sandwich with one egg and there
are none left over?
Draw muffins in groups of 8. Draw eggs in groups of 12.
Stop when you have drawn the same number of each.
There are 24 English muffins and 24 eggs.
So 3 packs of English muffins and 2 cartons of eggs are
needed.
5-1 Least Common Multiple
Check It Out: Example 1
Dog cookies come in packages of 6, and bones in bags of
9. If there are 18 dogs, what is the least number of
packages and bags needed so that each dog has a treat
box with one bone and one cookie and there are no
bones or cookies left over?
Draw cookies in groups of 6. Draw bones in groups of 9. Stop
when you have drawn the same number of each.
There are 18 dog cookies and 18 bones.
So 3 packages of dog cookies and 2 bags of bones are needed.
5-1 Least Common Multiple
The smallest number that is a multiple of
two or more numbers is the least
common multiple (LCM). In Additional
Example 1, the LCM of 8 and 12 is 24.
5-1 Least Common Multiple
Additional Example 2A: Using Multiples to Find
the LCM
Find the least common multiple (LCM).
Method 1: Use a number line.
3 and 4
Use a number line to skip count by 3 and 4.
0
2
4
6
8
10
12
The least common multiple (LCM) of 3 and 4 is 12.
5-1 Least Common Multiple
Additional Example 2B: Using Multiples to Find
the LCM
Find the least common multiple (LCM).
Method 2: Use a list.
4, 5, and 8
4: 4, 8, 12 , 16, 20, 24, 28, 32,
36, 40, 44, . . .
List multiples of 4, 5, and 8.
5: 5, 10, 15, 20, 25, 30, 35, 40,
45, . . .
8: 8, 16, 24, 32, 40, 48, . . .
LCM: 40
Find the smallest number
that is in all the lists.
5-1 Least Common Multiple
Remember!
The prime factorization of a number is
the number written as a product of its
prime factors.
5-1 Least Common Multiple
Additional Example 2C: Using Multiples to Find
the LCM
Find the least common multiple (LCM).
Method 3: Use prime factorization.
6 and 20
6=2•3
20 = 2
•2•5
Write the prime factorization of
each number.
Line up the common factors.
2•3 •2•5
To find the LCM, multiply one
2 • 3 • 2 • 5 = 60 number from each column.
LCM: 60
5-1 Least Common Multiple
Additional Example 2D: Using Multiples to Find
the LCM
Find the least common multiple (LCM).
15, 6, and 4
15 = 3 • 5
6=3•
2
4=
22
3 • 5 • 22
3 • 5 • 22 = 60
LCM: 60
Write the prime factorization of
each number in exponential form.
To find the LCM, multiply each
prime factor once with the
greatest exponent used in any of
the prime factorizations.
5-1 Least Common Multiple
Check It Out: Example 2A
Find the least common multiple (LCM).
Method 1: Use a number line.
2 and 3
Use a number line to skip count by 2 and 3.
0
1
2
3
4
5
6
The least common multiple (LCM) of 2 and 3 is 6.
5-1 Least Common Multiple
Check It Out: Example 2B
Find the least common multiple (LCM).
Method 2: Use a list.
3, 4, and 9
3: 3, 6, 9, 12, 15, 18, 21, 24,
27, 30, 33, 36, . . .
List multiples of 3, 4, and 9.
4: 4, 8, 12, 16, 20, 24, 28, 32,
36, …
Find the smallest number
that is in all the lists.
9: 9, 18, 27, 36, 45, . . .
The least common multiple of 3, 4, and 9 is 36.
5-1 Least Common Multiple
Check It Out: Example 2C
Find the least common multiple (LCM).
Method 3: Use prime factorization.
4 and 10
4=2•2
10 = 2
•5
Write the prime factorization of
each number.
Line up the common factors.
2•2 •5
2 • 2 • 5 = 20
LCM: 20
To find the LCM, multiply one
number from each column.
5-1 Least Common Multiple
Check It Out: Example 2D
Find the least common multiple (LCM).
12, 6, and 8
12 = 22 • 3
6=2• 3
Write the prime factorization of
each number in exponential form.
8 = 23
23 • 3
23 • 3 = 24
LCM: 24
To find the LCM, multiply each
prime factor once with the
greatest exponent used in any of
the prime factorizations.