Question

a. List all possible factorizations of 30 and 105 . b. How many different factorizations do each of these numbers have?

Short answer

Answer: There are 4 different factorizations for the number 30 and 4 different factorizations for the number 105.

Step-by-step solution

Understand the task

In this exercise, we need to find all possible factorizations of the numbers 30 and 105. Factorizations consist of pairs of factors that, when multiplied together, equal the given number. We'll list these factorizations and then determine the number of unique factorizations we found.

Factors of 30

To begin, let's find all the factors of 30. Factors are numbers that can be divided into our number without leaving a remainder. The factors of 30 are {}1, 2, 3, 5, 10, 15, 30. {}

Factorizations of 30

Now let's pair the factors with their corresponding counterpart to obtain the factorizations of 30. Here we go: 1 × 30 = 30 \\ 2 × 15 = 30 \\ 3 × 10 = 30 \\ 5 × 6 = 30 We have found 4 unique factorizations for 30: (1, 30) \\ (2, 15) \\ (3, 10) \\ (5, 6)

Factors of 105

Next, we need to find all the factors of 105. The factors of 105 are {}1, 3, 5, 7, 15, 21, 35, 105.{}

Factorizations of 105

Just like we did for 30, let's pair the factors to obtain the factorizations of 105: 1 × 105 = 105 \\ 3 × 35 = 105 \\ 5 × 21 = 105 \\ 7 × 15 = 105 We have found 4 unique factorizations for 105: (1, 105) \\ (3, 35) \\ (5, 21) \\ (7, 15)

Counting the factorizations

Now that we have listed the factorizations for both numbers, we can answer the second part of the question. We found 4 unique factorizations for 30, and 4 unique factorizations for 105. So, 30 has 4 different factorizations, and 105 also has 4 different factorizations.