What is the Least Common Multiple (LCM) of 112, 24, and 48?

The Least Common Multiple (LCM) of a set of numbers is the smallest multiple that is exactly divisible by each of the numbers in the set. We will use the prime factorization method to find the LCM of the given numbers 112, 24, and 48.

Prime Factorization of the Given Numbers

Let's start by breaking each number down into its prime factors.

Prime Factorization of 112:

112 can be factorized as:

{1122×2×2×2×7}

Prime Factorization of 24:

24 can be factorized as:

{242×2×2×3}

Prime Factorization of 48:

48 can be factorized as:

{482×2×2×2×3}

Finding the LCM Using Prime Factors

To find the LCM, we need to take the highest power of each prime factor that appears in the prime factorizations of the numbers.

Identifying the Highest Powers:

- For the prime factor 2, the highest power is 24 (from 112 and 48).

- For the prime factor 3, the highest power is 31 (from 24 and 48).

- For the prime factor 7, the highest power is 71 (from 112).

Calculating the LCM

Now, we multiply these highest powers together to find the LCM:

{LCM 24 × 31 × 71 336}

Conclusion

The LCM of 112, 24, and 48 is 336. Therefore, the final answer is:

{LCM 336}

Understanding how to find the LCM is helpful in solving various mathematical problems, such as adding or subtracting fractions with different denominators. If you have any further questions or need assistance with more examples, feel free to ask!