Question

Use the prime factorizations of these numbers to help you complete the statements: Prime factorization of 30: 2 * 3 * 5 Prime factorization of 45: 3 * 3 * 5 The factors that are common to both factorizations are: a) 2, 3, 5 b) 3, 5 c) 2, 3 d) 2, 5 The GCF of 30 and 45 is: a) 2 b) 3 c) 5 d) 15

Answer 1

The common factors of the prime factorizations of 30 and 45 are 3 and 5, making option (b) correct. The greatest common factor (GCF) of 30 and 45 is 15, which corresponds to option (d).

Step-by-step explanation:

When comparing the prime factorizations of two numbers to find their common factors and greatest common factor (GCF), we look for the prime numbers that appear in both factorizations. For the numbers 30 and 45, we have the following factorizations:

• Prime factorization of 30: 2 × 3 × 5
• Prime factorization of 45: 3 × 3 × 5

The factors common to both factorizations are 3 and 5, because these are the primes that appear in the factorizations of both numbers. Hence, the correct answer to which factors are common is option (b) 3, 5.

The GCF is found by multiplying the lowest powers of the common prime factors from the factorizations. Since 3 and 5 are the only prime numbers common to both factorizations, and both appear to the first power, the GCF of 30 and 45 is 3 × 5, which is 15. Therefore, the correct answer for the GCF of 30 and 45 is option (d) 15.

Answer 2

The factors that are common to both the prime factorizations of 30 and 45 are 3 and 5. The GCF of 30 and 45 is 15.

Step-by-step explanation:

The factors that are common to both the prime factorizations of 30 and 45 are 3 and 5. So, the answer to the first question is b) 3, 5.

To find the GCF (Greatest Common Factor) of 30 and 45, we look at the prime factors that are common to both numbers. In this case, the common prime factors are 3 and 5. The GCF is the product of these common prime factors, which is 3 * 5 = 15. So, the answer to the second question is d) 15.