How to find LCM by prime factorization?
Understand the Problem
The question is asking for a method to find the least common multiple (LCM) using prime factorization, which involves breaking down numbers into their prime factors and then using these factors to derive the LCM.
Answer
The LCM of 12 and 18 is 36.
Answer for screen readers
The least common multiple (LCM) of 12 and 18 is 36.
Steps to Solve
- Prime Factorization of Each Number
First, break down each number into its prime factors. For example, if we want to find the LCM of 12 and 18:
- The prime factorization of 12 is ( 12 = 2^2 \times 3^1 ).
- The prime factorization of 18 is ( 18 = 2^1 \times 3^2 ).
- Identify the Highest Power of Each Prime
Next, identify each unique prime factor from both factorizations and take the highest power for each:
- For prime factor 2: the highest power is ( 2^2 ) (from 12).
- For prime factor 3: the highest power is ( 3^2 ) (from 18).
- Multiply the Highest Powers Together
Now, multiply these highest powers together to find the LCM:
$$ LCM = 2^2 \times 3^2 $$
- Calculate the LCM
Finally, carry out the multiplication:
$$ LCM = 4 \times 9 = 36 $$
The least common multiple (LCM) of 12 and 18 is 36.
More Information
The least common multiple is the smallest number that is a multiple of both numbers. It is useful in various mathematical applications including finding common denominators in fractions.
Tips
- Not finding all prime factors correctly, which can result in an incorrect LCM. Always double-check your factorization.
- Forgetting to include all unique prime factors when determining the highest power.
