What is the LCM of 12 and 28?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 12 and 28. To solve this, we will first find the prime factorization of both numbers and then use these factors to determine the LCM.
Answer
The least common multiple of 12 and 28 is $84$.
Answer for screen readers
The least common multiple (LCM) of 12 and 28 is $84$.
Steps to Solve
- Prime Factorization of 12
Start by finding the prime factorization of the number 12. The prime factors of 12 are: $$ 12 = 2^2 \times 3^1 $$
- Prime Factorization of 28
Next, find the prime factorization of the number 28. The prime factors of 28 are: $$ 28 = 2^2 \times 7^1 $$
- Identify the Highest Powers of Each Factor
List all prime factors from both factorizations and take the highest power of each:
- For the factor 2, the highest power is $2^2$ (from both 12 and 28).
- For the factor 3, the highest power is $3^1$ (from 12).
- For the factor 7, the highest power is $7^1$ (from 28).
- Calculate the LCM Using the Highest Powers
Multiply the highest powers of all prime factors together to find the LCM: $$ LCM = 2^2 \times 3^1 \times 7^1 $$
- Perform the Multiplication
Now calculate the numerical value of the LCM: $$ LCM = 4 \times 3 \times 7 $$
Calculate step-by-step: First, calculate $4 \times 3 = 12$. Then, calculate $12 \times 7 = 84$.
The least common multiple (LCM) of 12 and 28 is $84$.
More Information
The LCM is useful in various applications, such as finding common denominators in fractions or synchronizing cycles in problems involving repeated events.
Tips
- Forgetting to include all prime factors when determining the highest powers can lead to an incorrect LCM.
- Confusing the LCM with the greatest common divisor (GCD), which is a different concept.