What is the LCM of 48 and 32?
Understand the Problem
The question is asking us to find the least common multiple (LCM) of the numbers 48 and 32. The LCM is the smallest number that is a multiple of both numbers.
Answer
The least common multiple of 48 and 32 is $96$.
Answer for screen readers
The least common multiple (LCM) of 48 and 32 is $96$.
Steps to Solve
- Find the prime factorization of each number
To find the LCM, we start with the prime factorization of both 48 and 32.
-
The prime factorization of 48 is: $$ 48 = 2^4 \times 3^1 $$
-
The prime factorization of 32 is: $$ 32 = 2^5 $$
- Identify the highest powers of each prime factor
Next, we take the highest power of each prime factor from both factorizations:
- For the prime factor 2, the highest power is $2^5$.
- For the prime factor 3, the highest power (which only appears in the factorization of 48) is $3^1$.
- Multiply the highest powers together
Now we can multiply these highest powers to find the LCM:
$$ LCM = 2^5 \times 3^1 $$
- Calculate the final result
Now we calculate the product:
$$ LCM = 32 \times 3 = 96 $$
The least common multiple (LCM) of 48 and 32 is $96$.
More Information
The least common multiple is useful in various mathematical applications, particularly in solving problems involving fractions and finding common denominators. In this case, $96$ is the smallest number that both $48$ and $32$ can divide evenly into.
Tips
- A common mistake is to simply multiply the two numbers together. While this gives a common multiple, it doesn't guarantee that it's the least common one.
- Forgetting to take the highest power of each prime factor can lead to incorrect results.
