How to find least common multiple with prime factorization?
Understand the Problem
The question is asking for a method to calculate the least common multiple (LCM) of numbers using their prime factorization. This involves finding the prime factors of each number and then using these factors to determine the LCM.
Answer
The LCM is $60$.
Answer for screen readers
The least common multiple (LCM) is $60$.
Steps to Solve
- Find the prime factorization of each number
For each number, we will break it down into its prime factors. For example,
- The prime factorization of 12 is: $$ 12 = 2^2 \times 3^1 $$
- The prime factorization of 15 is: $$ 15 = 3^1 \times 5^1 $$
- List all prime factors with their highest powers
Next, we will list all the prime factors found from each number and identify the highest power of each factor.
From our example:
- Prime factor 2 has the highest power of $2^2$ (from 12).
- Prime factor 3 has the highest power of $3^1$ (from both).
- Prime factor 5 has the highest power of $5^1$ (from 15).
- Multiply the highest powers of all prime factors
Now, we will multiply the highest powers of all prime factors to get the LCM.
Using our previous results: $$ LCM = 2^2 \times 3^1 \times 5^1 $$
- Calculate the final result
Now we perform the multiplication: $$ LCM = 4 \times 3 \times 5 = 60 $$
The least common multiple (LCM) is $60$.
More Information
The LCM is the smallest number that both original numbers can divide without leaving a remainder. It is useful in various applications, including finding common denominators in fractions.
Tips
- Forgetting to take the highest power of each prime factor when listing them.
- Miscalculating the multiplication of the highest powers, leading to an incorrect LCM.
