What is the lowest common multiple of 48 and 60?

Steps to Solve

1. Prime Factorization of 48

First, we find the prime factorization of 48.

To do this, we divide 48 by the smallest prime number until we cannot divide anymore:

$$ 48 \div 2 = 24 \ 24 \div 2 = 12 \ 12 \div 2 = 6 \ 6 \div 2 = 3 \ 3 \div 3 = 1 $$

Thus, the prime factorization of 48 is:

$$ 48 = 2^4 \times 3^1 $$

2. Prime Factorization of 60

Next, we find the prime factorization of 60 in the same manner:

$$ 60 \div 2 = 30 \ 30 \div 2 = 15 \ 15 \div 3 = 5 \ 5 \div 5 = 1 $$

So, the prime factorization of 60 is:

$$ 60 = 2^2 \times 3^1 \times 5^1 $$

3. Identifying Maximum Powers of Each Prime Factor

Now, we compile the factors.

For the primes we found, we take the highest power from each number:

• For $2$: max is $2^4$ (from 48)
• For $3$: max is $3^1$ (common in both)
• For $5$: max is $5^1$ (from 60)

4. Calculate the LCM

We multiply these maximum powers to find the LCM:

$$ \text{LCM} = 2^4 \times 3^1 \times 5^1 $$

Now we compute this step-by-step:

$$ 2^4 = 16 \ 3^1 = 3 \ 5^1 = 5 $$

Now we multiply these results together:

$$ 16 \times 3 = 48 \ 48 \times 5 = 240 $$

So, the LCM of 48 and 60 is 240.