LCM of 48 and 60
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 48 and 60. To solve this, we will first find the prime factorizations of both numbers and then use those to calculate the LCM.
Answer
The least common multiple (LCM) of 48 and 60 is $240$.
Answer for screen readers
The least common multiple (LCM) of 48 and 60 is $240$.
Steps to Solve
- Find the prime factorization of 48
To factor 48 into its prime factors, we divide by the smallest prime number.
$$ 48 = 2 \times 24 $$ $$ 24 = 2 \times 12 $$ $$ 12 = 2 \times 6 $$ $$ 6 = 2 \times 3 $$
So, the prime factorization of 48 is:
$$ 48 = 2^4 \times 3^1 $$
- Find the prime factorization of 60
Similarly, for 60, we will also divide by the smallest prime number.
$$ 60 = 2 \times 30 $$ $$ 30 = 2 \times 15 $$ $$ 15 = 3 \times 5 $$
Thus, the prime factorization of 60 is:
$$ 60 = 2^2 \times 3^1 \times 5^1 $$
- Identify the highest powers of the prime factors
Next, we take the highest power of each prime number that appears in either factorization.
- For the prime factor 2, the highest power is $2^4$ (from 48).
- For the prime factor 3, the highest power is $3^1$ (from both).
- For the prime factor 5, the highest power is $5^1$ (from 60).
- Calculate the LCM using the highest powers
To find the LCM, we multiply these highest powers together:
$$ \text{LCM} = 2^4 \times 3^1 \times 5^1 $$
Calculating this step by step:
$$ 2^4 = 16 $$ $$ 3^1 = 3 $$ $$ 5^1 = 5 $$
Now, multiply these together:
$$ \text{LCM} = 16 \times 3 \times 5 $$
- Perform the final multiplication
First, we calculate:
$$ 16 \times 3 = 48 $$
Then, multiply by 5:
$$ 48 \times 5 = 240 $$
Thus, the least common multiple of 48 and 60 is 240.
The least common multiple (LCM) of 48 and 60 is $240$.
More Information
The LCM is useful in various mathematical applications, such as finding a common denominator in fractions or solving problems involving periodic events.
Tips
- Forgetting to include all prime factors when identifying the highest powers can lead to an incorrect LCM.
- Not checking that you have the correct prime factorization for each number.