What is the least common multiple of 48 and 24?

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 48 and 24. The LCM is the smallest number that is a multiple of both 48 and 24.

Answer

$48$

Steps to Solve

Identify the prime factors of each number

To find the LCM, we first determine the prime factorization of each number.

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

For 24: $$ 24 = 2^3 \times 3^1 $$

Choose the highest power of each prime factor

Next, we take the highest power of each prime factor we find in the factorizations.

For prime factor $2$: the highest power is $2^4$ (from 48).
For prime factor $3$: the highest power is $3^1$ (both have the same power).

Multiply the highest powers together

Now, we multiply these highest powers to find the LCM.

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

Calculating this gives us:

$$ LCM = 16 \times 3 = 48 $$

The least common multiple (LCM) of 48 and 24 is $48$.

More Information

The LCM of two numbers is useful in various mathematical applications, including adding fractions with different denominators. The LCM represents the smallest value that can be evenly divided by both numbers.

Tips

A common mistake is forgetting to take the highest power of the prime factors. Always make sure to check both factorizations.
Another mistake is to simply multiply the two numbers to find the LCM, which does not guarantee the smallest common multiple.

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