Understanding the Least Common Multiple (LCM) in Fractions

In mathematics, particularly when dealing with fractions, the Least Common Multiple (LCM) of the denominators plays a crucial role. This is often necessary for operations such as adding or subtracting fractions. This article will guide you through the process of finding the LCM, specifically for the fractions 1/6 and 5/8.

Step-by-Step Process to Find the LCM of 6 and 8

To find the LCM of the denominators 6 and 8, we first identify their prime factorizations:

The prime factorization of 6 is 2 × 3. The prime factorization of 8 is 2^3.

The next step is to identify the highest power of each prime factor involved.

For the prime factor 2, the highest power is 2^3 (from 8). For the prime factor 3, the highest power is 3^1 (from 6).

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

LCM 2^3 × 3^1 8 × 3 24.

Therefore, the least common multiple of the denominators 6 and 8 is 24.

Application in Adding Fractions

Once we have the LCM, we can use it to express both fractions with a common denominator. This allows us to add or subtract the fractions more easily.

Given the fractions 1/6 and 5/8, we follow these steps:

Express 1/6 and 5/8 using the LCM (24) as the common denominator. 1/6 4/24 5/8 15/24

Add the fractions: 4/24 15/24 19/24.

Since 19 and 24 have no common factors other than 1, the fraction 19/24 is in its simplest form.

Alternative Method: Using Factors

An alternative method involves checking multiples of the larger number until it is divisible by the smaller number. For 6 and 8, we start with the larger number 8 and check its multiples:

8 × 1 8 (not divisible by 6) 8 × 2 16 (not divisible by 6) 8 × 3 24 (divisible by 6)

Thus, the least common multiple of 6 and 8 is 24.

Conclusion

The least common multiple of the denominators 6 and 8 is 24. This method of finding the LCM and using it to express fractions with a common denominator is key to performing operations on fractions accurately and efficiently.