how can you use prime factorization to determine the LCM and GCF of two numbers?

Write down the prime factors of each.

Look at all the factors, of both numbers, together.

The LCM will be the smallest set of this list from which you can make either of the two original numbers.

Example:

30 = 2 * 3 * 5
24 = 2 * 2 * 2 * 3

So we have 2, 2, 2, 2, 3, 3, 5 to choose from, and we need to choose a set from that that will make either of our numbers. That set is: 2 * 2 * 2 * 3 * 5, which is 120.

Now, for the GCF, we need to choose as many of the factors as both number have in common:

30 = 2 * 3 * 5
24 = 2 * 2 * 2 * 3

2 and 3 are in both (only one two, since 30 has only one two), so 2 * 3 = 6 is the GCF.

Here is how you can use prime factorization to determine the Least Common Multiple (LCM) and Greatest Common Factor (GCF) of two numbers:

1. Start by finding the prime factorization of each number separately. Prime factorization is the process of breaking down a number into its prime factors.

2. To find the prime factorization of a number, you divide it by the smallest prime number possible, and continue dividing until the quotient is no longer divisible evenly by that prime number. Repeat this process with the quotient until the quotient itself is a prime number.

3. Once you have the prime factorization of both numbers, you can use it to determine the LCM and GCF.

4. For the LCM, take all the prime factors of both numbers, including the repeated factors, and multiply them together. This will give you the LCM of the two numbers.

5. For the GCF, identify the common prime factors between the two numbers, and multiply them together. If two numbers do not have any common prime factors, then the GCF is 1.

Let's go through an example to illustrate the process:

Example: Find the LCM and GCF of 20 and 36.

1. Prime factorization of 20: 2 x 2 x 5 = 2^2 x 5
Prime factorization of 36: 2 x 2 x 3 x 3 = 2^2 x 3^2

2. LCM: Take all the prime factors, including the repeated factors, and multiply them together.
LCM = 2^2 x 3^2 x 5 = 4 x 9 x 5 = 180

3. GCF: Identify the common prime factors between the two numbers, and multiply them together.
GCF = 2^2 = 4

Therefore, the LCM of 20 and 36 is 180, and the GCF is 4.

To use prime factorization to determine the LCM (Least Common Multiple) and GCF (Greatest Common Factor) of two numbers, you can follow these steps:

1. Start by finding the prime factorization of both numbers.
- Prime factorization entails expressing a number as the product of its prime factors.
- For example, let's consider the numbers 24 and 36:
- Prime factorization of 24: 2 * 2 * 2 * 3 = 2^3 * 3
- Prime factorization of 36: 2 * 2 * 3 * 3 = 2^2 * 3^2

2. To determine the GCF, identify the common prime factors with their lowest exponents.
- Take the highest exponent for each common prime factor in both numbers.
- In our example, the common prime factors are 2 and 3.
- The GCF is the product of these, so GCF(24, 36) = 2^2 * 3 = 12.

3. To find the LCM, identify all the unique prime factors with their highest exponents.
- Take the highest exponent for each prime factor found in both numbers.
- Include any additional prime factors that are unique to either number.
- In our example, the unique prime factors are 2 (highest exponent 3) and 3 (highest exponent 2).
- The LCM is the product of these, so LCM(24, 36) = 2^3 * 3^2 = 72.

By utilizing the prime factorizations of the numbers, you can determine the GCF and LCM accurately.