Question

Use a factor tree to write the prime factorization of the numbers. Then find the GCF and the LCM of the numbers:

a) 45, 150
b) 68, 102

Answer 1

To find the prime factorization of the given numbers, we can use a factor tree. For a) 45 and 150, the prime factorization of 45 is 3 x 3 x 5 and the prime factorization of 150 is 2 x 3 x 5 x 5. The GCF is 3 and the LCM is 450. For b) 68 and 102, the prime factorization of 68 is 2 x 2 x 17 and the prime factorization of 102 is 2 x 3 x 17. The GCF is 34 and the LCM is 204.

Step-by-step explanation:

To find the prime factorization of a number, we can use a factor tree.

Let's start with a) 45:

1. Divide 45 by the smallest prime number, which is 3. We get 15.
2. Divide 15 by 3 again to get 5.
3. Since 5 is a prime number, the prime factorization of 45 is 3 x 3 x 5.

Now let's find the GCF and LCM:

• For GCF, we look for the highest common factor among the prime factors. In this case, the GCF is 3.
• For LCM, we multiply the highest power of each prime factor. So the LCM is 3 x 3 x 5 = 45.

Now let's move on to b) 68 and 102:

1. Divide 68 by the smallest prime number, which is 2. We get 34.
2. Divide 34 by 2 again to get 17.
3. Since 17 is a prime number, the prime factorization of 68 is 2 x 2 x 17.
4. Next, divide 102 by 2 to get 51.
5. Divide 51 by 3 to get 17.
6. Since 17 is a prime number, the prime factorization of 102 is 2 x 3 x 17.

For GCF, we look for the highest common factor among the prime factors. In this case, the GCF is 2 x 17 = 34.

For LCM, we multiply the highest power of each prime factor. So the LCM is 2 x 2 x 3 x 17 = 204.