Question

$$\begin{array}{cc}\text { Quantity } \mathbf{A} & \text { Quantity } \mathbf{B} \\ \text {The number of distinct positive integer factors of 96 } & \text {The number of distinct positive integer factors of 72} \end{array}$$ A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

Short answer

C. The two quantities are equal.

Step-by-step solution

Prime Factorization of 96

Perform the prime factorization of 96.\[ 96 = 2^5 \times 3^1 \]

Find Number of Factors for 96

Using the prime factors of 96, calculate the number of distinct positive integer factors. The formula to find the number of factors is \((e_1+1) \times (e_2+1) \times ... \times (e_n+1)\), where \( e_1, e_2, ..., e_n \) are the powers of the prime factors.\[ \text{For } 96 = 2^5 \times 3^1: (5+1) \times (1+1) = 6 \times 2 = 12 \]

Prime Factorization of 72

Perform the prime factorization of 72.\[ 72 = 2^3 \times 3^2 \]

Find Number of Factors for 72

Using the prime factors of 72, calculate the number of distinct positive integer factors. Again use the formula \((e_1+1) \times (e_2+1) \times ... \times (e_n+1)\).\[ \text{For } 72 = 2^3 \times 3^2: (3+1) \times (2+1) = 4 \times 3 = 12 \]

Compare the Quantities

Both Quantity A (number of factors of 96) and Quantity B (number of factors of 72) are equal to 12.

Key concepts

Prime Factorization

Understanding prime factorization is essential for solving many mathematical problems. Prime factorization involves breaking down a number into its smallest prime components. A prime number is a number greater than 1 that has no other divisors other than 1 and itself. For instance, 2, 3, 5, and 7 are prime numbers.

In the problem, we first factorize 96 and 72 into their prime factors:

• For 96, the prime factorization is: 96 = 2^5 × 3^1.
• For 72, the prime factorization is: 72 = 2^3 × 3^2.

This means that 96 can be expressed as the product of 2 raised to the 5th power and 3 raised to the 1st power. Similarly, 72 can be written as the product of 2 raised to the 3rd power and 3 raised to the 2nd power. These factorizations help in further calculations, especially when determining the number of factors each number has.

Number of Factors Formula

Once we have the prime factorization of a number, we can easily find the number of its positive integer factors using a specific formula. The formula for finding the number of distinct positive factors of a number is: \( (e_1+1) \times (e_2+1) \times \text{...} \times (e_n+1) \), where \( e_1, e_2,..., e_n \) are the exponents in the prime factorization of the number.

Let's apply this formula:

• For 96: The prime factorization is 2^5 × 3^1. So the number of positive integer factors is \( (5+1) \times (1+1) = 6 \times 2 = 12 \).
• For 72: The prime factorization is 2^3 × 3^2. Hence, the number of positive integer factors is \( (3+1) \times (2+1) = 4 \times 3 = 12 \).

This calculation shows that both numbers, 96 and 72, have exactly 12 distinct positive integer factors.

Positive Integer Factors

Positive integer factors of a number are all the integers that can divide the number without leaving a remainder. For example, the positive factors of 6 are 1, 2, 3, and 6. Understanding prime factorization helps us determine the total number of these factors systematically.

Let's consider how prime factorization aids in finding these factors: By knowing the prime factors and their powers, we can systematically list out all the products of these factors that divide the number.

• For 96: The prime factorization is 2^5 × 3^1. All the combinations of these factors (like \(2^0 \times 3^0, 2^1 \times 3^0, ...2^5 \times 3^1\)) will provide the actual positive factors. There are 12 such combinations, which correspond to its 12 positive integer factors.
• For 72: The prime factorization is 2^3 × 3^2. Similarly, all possible products of these factor combinations give us the 12 distinct positive factors of 72.

So, whenever you have to find the number of positive integer factors of any number, first perform its prime factorization, then use the number of factors formula to get an accurate count.