Calculating the Least Common Multiple (LCM) Using Prime Factorization

The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of all the numbers in the set. One common method to find the LCM is through prime factorization. This article will guide you through the process of finding the LCM of 45, 75, and 60 using the prime factorization method, explaining each step in detail.

Prime Factorization

Let's begin by determining the prime factorization of each number.

Prime Factorization of 45

45 32 × 51

Prime Factorization of 75

75 31 × 52

Prime Factorization of 60

60 22 × 31 × 51

Identifying the Highest Powers of Each Prime Factor

The next step is to identify the highest power of each prime factor found in the factorizations.

Highest power of 2: 22 from 60.

Highest power of 3: 32 from 45.

Highest power of 5: 52 from 75.

Combining the Highest Powers to Calculate the LCM

Now, combine the highest powers of each prime factor to calculate the LCM.

LCM 22 × 32 × 52

Calculate each component:

22 4

32 9

52 25

Multiply the components together:

4 × 9 36

36 × 25 900

Thus, the LCM of 45, 75, and 60 is 900.

Alternative Methods for Finding LCM

There are other methods for finding the LCM, such as the Vedic Mathematics concept. However, the prime factorization method is straightforward and widely used. An alternative, as seen in the observation below, is to use the highest common factor (HCF) of the numbers to find the LCM.

Note: The HCF of 45, 75, and 60 is 15, but this method does not apply here as 15 is not a factor of all three numbers.

Another approach is to directly multiply the highest powers of the prime factors:

Prime Factorization of 45, 60, and 75 is:

45 3 × 3 × 5 32 × 51

75 3 × 5 × 5 31 × 52

60 2 × 2 × 3 × 5 22 × 31 × 51

The LCM can be obtained by multiplying prime factors raised to their respective highest power:

LCM 22 × 32 × 52 900

This confirms the result obtained through the prime factorization method.

Conclusion

The LCM of 45, 75, and 60 is 900, which can be found by identifying the highest powers of the prime factors and then multiplying these powers together. This method is reliable and effective, making it a standard approach in mathematics.