Question

Quantity \(A\) The number of distinct positive integer factors of 96 Quantity \(B\) The number of distinct positive integer factors of 72 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

The two quantities are equal.

Step-by-step solution

- Prime Factorization of 96

Determine the prime factorization of 96. Begin by dividing by the smallest prime number:96 ÷ 2 = 4848 ÷ 2 = 2424 ÷ 2 = 1212 ÷ 2 = 66 ÷ 2 = 33 ÷ 3 = 1Thus, the prime factorization of 96 is 96 = 2^5 × 3^1

- Number of Distinct Positive Integer Factors of 96

Use the formula for finding the number of factors: if a number is of the form n = p^a × q^b then the number of distinct positive integer factors is (a+1)(b+1) For 96, a = 5 (for 2), b = 1 (for 3) (5+1)(1+1) = 6×2 = 12 Thus, the number of distinct positive integer factors of 96 is 12.

- Prime Factorization of 72

Determine the prime factorization of 72. Begin by dividing by the smallest prime number:72 ÷ 2 = 3636 ÷ 2 = 1818 ÷ 2 = 99 ÷ 3 = 33 ÷ 3 = 1Thus, the prime factorization of 72 is 72 = 2^3 × 3^2

- Number of Distinct Positive Integer Factors of 72

Use the same formula for finding the number of factors:For 72, a = 3 (for 2), b = 2 (for 3) (3+1)(2+1) = 4×3 = 12 Thus, the number of distinct positive integer factors of 72 is 12.

- Compare the Quantities

Compare the number of distinct positive integer factors of 96 and 72:Both quantities have 12 distinct positive integer factors.

Key concepts

prime factorization

Prime factorization is a fundamental concept in number theory. It involves breaking down a composite number into a product of prime numbers. Prime numbers are those that have exactly two distinct positive divisors: 1 and themselves. For example, to find the prime factorization of 96, we start by dividing it by the smallest prime number, which is 2:
96 ÷ 2 = 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
Since 3 is the next smallest prime, we divide once more:
3 ÷ 3 = 1
Hence, the prime factorization of 96 is \(2^5 \times 3^1\).
This process breaks down a number into factors that are primes, making it easier to explore other properties of the number, such as finding its divisors.

integer factors

Integer factors of a number are the whole numbers that divide it exactly without leaving a remainder. To find the number of distinct positive integer factors of a number, we can use its prime factorization. If a number n is represented as \(p^a \times q^b\), where p and q are prime factors, the number of distinct positive integer factors is given by \((a+1)(b+1)\).
For instance, given the prime factorization of 96 as \(2^5 \times 3^1\),we can determine the number of integer factors by:
\((5+1)(1+1) = 6 \times 2 = 12\).
This means 96 has 12 distinct positive integer factors. Similarly, for 72 with its prime factorization \(2^3 \times 3^2\),we find the number of factors as:
\((3+1)(2+1) = 4 \times 3 = 12\).
Here too, 72 has 12 distinct positive integer factors.

number theory

Number theory is a branch of mathematics that deals with properties and relationships of numbers, particularly integers. A central aspect of number theory is exploring how numbers can be factored and how those factors interact. Prime factorization is a key tool in number theory because it breaks numbers down to their most basic building blocks, the prime numbers. This decomposition helps in understanding the structure and properties of numbers better.
Number theory also includes the study of integers’ relationships, such as divisibility, common factors, and modular arithmetic. Concepts like prime numbers, divisors, and prime factorization play crucial roles in various problems and applications, including cryptography and coding theory.

educational math problems

Educational math problems designed to teach prime factorization and integer factors help students not only understand number properties but also enhance their problem-solving skills. These types of problems often ask students to break down numbers into their prime factors, find the number of distinct factors, or compare properties of different numbers. Such exercises develop critical thinking and a deeper understanding of number theory.
For instance, the exercise provided asks students to compare the number of distinct positive integer factors of 96 and 72. This problem enforces concepts like prime factorization and the calculation of factors. It demonstrates that despite different primes in their composition, both numbers have the same number of factors. Educational math problems like these can strengthen foundational math skills and prepare students for more advanced topics.