LCM of 48 and 60

Steps to Solve

1. 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:

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

2. 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:

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

3. 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).

4. 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:

$$ \text{LCM} = 16 \times 3 \times 5 $$

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.