How to Find the Least Common Multiple of 20 and 25 Using Prime Factorization

When faced with the task of finding the least common multiple (LCM) of two numbers, particularly 20 and 25, understanding how to use prime factorization can be very helpful. In this article, we will walk through the process of determining the LCM of 20 and 25 by prime factorization and explore different methods to find the LCM.

Prime Factorization Method

The prime factorization method is a systematic and efficient way to find the LCM. By breaking down each number into its prime factors, we can easily identify the highest powers of each prime and then multiply them.

Step 1: Prime Factorization

To start, we need to find the prime factorization of each number:

20 2^2 × 5^1 25 5^2

Step 2: Identify the Highest Powers of Each Prime

Next, we identify the highest powers of each prime number from the factorizations:

For the prime number 2: the highest power is 2^2 from 20. For the prime number 5: the highest power is 5^2 from 25.

Step 3: Multiply These Together

To find the LCM, we multiply these highest powers together:

LCM 2^2 × 5^2 4 × 25 100

Alternative Methods to Find the LCM

There are other methods to find the LCM of 20 and 25, which we will explore in the following sections.

Method A: Listing Multiples

A common method is to list the multiples of each number and identify the smallest common multiple. Here's how it works:

Multiples of 20: 20, 40, 60, 80, 100, 120 Multiples of 25: 25, 50, 75, 100, 125

By comparing the lists, we can see that the least common multiple is 100.

Method B: Using Prime Factorization

Another method involves listing the prime factorization of each number and then multiplying the largest power of each prime. Starting with the prime factorization:

20 2^2 × 5^1 25 5^2

We need the largest power of each prime from both factorizations:

For the prime number 2, the highest power is 2^2. For the prime number 5, the highest power is 5^2.

Multiplying these highest powers together gives us the LCM:

LCM 2^2 × 5^2 4 × 25 100

Conclusion

The least common multiple of 20 and 25 is 100. Whether you use the prime factorization method, listing multiples, or any other mathematical approach, understanding these techniques can be invaluable for solving similar problems involving LCM.

In summary, the LCM of 20 and 25 is 100. The multiples of 20 and 25 intersect at 100, making it the least positive common multiple of both numbers. Here's a brief recap of the methods used:

Prime Factorization Method: 20 22 × 51 and 25 52; thus, LCM 22 × 52 100. Listing Multiples Method: Finding the first common multiple in the lists of multiples of 20 and 25. Prime Factorization and Multiplication: Using the highest power of each prime from both factorizations.

With practice, you'll be able to find the LCM of any two numbers more efficiently. Happy calculating!