What is the least common multiple of 48 and 60?

Understand the Problem

The question is asking for the least common multiple (LCM) of two numbers, 48 and 60. To find this, we will determine the smallest positive integer that is a multiple of both numbers.

Answer

The least common multiple of 48 and 60 is $240$.

Steps to Solve

Find the prime factorization of each number

To find the LCM, we first need to break down each number into its prime factors:

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

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

Identify the highest powers of all prime factors

Next, we will list all the prime factors involved and take the highest power of each prime factor from both factorizations:

For 2, the highest power is $2^4$ (from 48).
For 3, the highest power is $3^1$ (common in both).
For 5, the highest power is $5^1$ (from 60).

Multiply these highest powers together

Now we will multiply these highest powers to find the LCM:

$$ LCM = 2^4 \times 3^1 \times 5^1 $$

Calculating this out:

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

Thus, $$ LCM = 16 \times 3 \times 5 $$

Calculate the final result

Now, we will do the multiplication step-by-step:

First, multiply $16 \times 3$:

$$ 16 \times 3 = 48 $$

Now, multiply that result by $5$:

$$ 48 \times 5 = 240 $$

So, we find that:

$$ LCM(48, 60) = 240 $$

The least common multiple of 48 and 60 is $240$.

More Information

The least common multiple is useful in a variety of applications, such as finding a common denominator or solving problems involving synchronized events. The prime factorization method is one of the clearest ways to find the LCM.

Tips

Forgetting to consider all prime factors: Always ensure that you identify the highest powers of all unique prime factors across both numbers.
Miscalculating the multiplication steps: Double-check each multiplication step to avoid errors.

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