lcm of 60 and 48

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 60 and 48. To solve this, we will find the multiples of both numbers and identify the smallest multiple that they both share.

Answer

$240$

Steps to Solve

Find the prime factorization of each number

To find the least common multiple (LCM), we first need to find the prime factorization of both numbers.

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

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

Identify the highest powers of all prime factors

Next, we identify all the prime factors from both numbers and take the highest power of each prime factor.

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

Multiply the highest powers together

Now we multiply the highest powers of each prime factor to find the LCM:

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

Calculating this gives:

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

Calculate the LCM

Now perform the multiplication step by step:

First, calculate $16 \times 3 = 48$.
Next, multiply $48 \times 5 = 240$.

Therefore, the LCM of 60 and 48 is:

$$ LCM = 240 $$

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

More Information

The least common multiple is significant in various applications, including finding common denominators in fractions and solving problems involving multiples of numbers, like scheduling events.

Tips

Forgetting to take the highest power of each prime factor can lead to an incorrect LCM. Always double-check the prime factorizations.
Using addition instead of multiplication when working with prime factors is a common error. Remember, LCM involves multiplication of the highest powers.

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