Download M84 Act 13 LCM

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Math 84
Activity # 13
Your Name: ___________________________
“Least Common Multiple”
Find the least common multiple of 24 and 30.
Method 1: Listing multiples
List the multiples of 24:
24, __, __, 96, ___, 144…
List the multiples of 30:
30, __, 90, ___, 150…
The Least common multiple of 24 and 30 is ____.
Method 2: Using prime factorization and exponents.
2
24
30
3 10
2 5
12
3 4
2 2
Prime factorization is
2⋅2⋅2⋅3
Prime factorization is 2 ⋅ 3 ⋅ 5
The Fundamental Theorem of Arithmetic states that every integer greater than
1 is either prime or may be uniquely expressed as the product of prime numbers.
We use this theorem to find the least common multiple (LCM).
1) Use factor trees to find the unique prime factorization of each number (as
seen above)
2) Then use exponential notation to represent the prime factorization.
The prime factorization of 24 is 2 ⋅ 2 ⋅ 2 ⋅ 3 = 23 ⋅ 31
The prime factorization of 30 is 2 ⋅ 3 ⋅ 5 = 21 ⋅ 31 ⋅ 51
Observe we place the bases in ascending order (smallest to largest).
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3) Write down all of the different bases exactly once. 2 ⋅ 3 ⋅ 5 (we do not
repeat the 2or 3)
4) Write down the largest exponent per base 2 3 ⋅ 31 ⋅ 51
5) Multiply out the product 23 ⋅ 31 ⋅ 51 = 8 ⋅ 3 ⋅ 5 = 8 ⋅15 = 120 .
Therefore, 120 is the least common multiple (LCM) of 24 and 30. It is the
smallest number that both 24 and 30 will divide into leaving no remainder.
Method 3: Using prime factorization
To find the least common multiple of 24 and 30, set up a table, and in the row next
to each number list its prime factorization by:
First, insert the smallest prime number from the factoring
24:
2
30:
2
2
2
LCM
Second, insert the next smallest prime number from the factoring, BUT start in
the column that doesn’t include the previous prime numbers.
24:
2
30:
2
LCM
2
2
3
3
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Third, then repeat the above step until all prime factorization for each number is
included in its row.
24:
2
30:
2
2
2
3
3
5
LCM
Fourth, the LCM will then be calculated in the bottom row by bringing down each
number to the bottom row and then finding their product (multiplying them out).
24:
2
30:
2
LCM
2
2
2
2
3
2
3
5
3
5
The LCM is the product of all the primes in the boxes labeled LCM which is
2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 120 .
Problem 1: Find the LCM of 18 and 48.
Find the prime factorization of 18 and 48 below.
18
a. Prime factorization is
_________.
48
b. Prime factorization is
_________.
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Find the LCM of 18 and 48. Insert all the prime numbers in the corresponding
boxes below.
18:
48:
LCM
c. What is the LCM of 18 and 48? = ______
Problem 2: Use any method to find the least common multiple for a – h, below..
a) Find the LCM of 12 and 40.
b) Find the LCM of 14 and 18.
c) Find the LCM of 20 and 36.
d) Find the LCM of 15 and 35.
e) Find the LCM of 14, 18 and 36.
f) Find the LCM of 8, 16, 38.
g) Find the LCM of 120 and 144.
h) Find the LCM of 18 and 56.