What Times What Is 84

What Times What Is 84? Exploring the Factors and Prime Factorization of 84

Finding the numbers that multiply to equal 84 might seem like a simple arithmetic problem, but it opens a door to deeper mathematical concepts like factors, prime factorization, and even the beginnings of algebra. This comprehensive guide will explore all aspects of this seemingly simple question, providing you with a complete understanding of the factors of 84 and the methods used to find them. We'll delve into the process, explain the underlying mathematical principles, and offer different approaches to solve similar problems.

Introduction: Understanding Factors and Multiples

Before we dive into finding the numbers that multiply to 84, let's establish some fundamental definitions. A factor of a number is a whole number that divides evenly into that number without leaving a remainder. For example, 2 is a factor of 8 because 8 divided by 2 is 4 (with no remainder). Conversely, a multiple of a number is the result of multiplying that number by another whole number. For instance, 8 is a multiple of 2 because 2 x 4 = 8.

Our goal is to find all the pairs of factors that, when multiplied together, result in 84. This involves identifying all the numbers that divide evenly into 84.

Method 1: Systematic Listing of Factors

The most straightforward approach to find the factors of 84 is to systematically list all the possible pairs of numbers that multiply to 84. We can start by considering the smallest factors and work our way up:

• 1 x 84: The smallest factor is always 1, and its pair is the number itself.
• 2 x 42: 84 is an even number, so it's divisible by 2.
• 3 x 28: The sum of the digits of 84 (8 + 4 = 12) is divisible by 3, indicating that 84 is divisible by 3.
• 4 x 21: 84 is divisible by 4 since it's divisible by 2 twice.
• 6 x 14: 84 is divisible by both 2 and 3, therefore it's divisible by 6.
• 7 x 12: This is a bit less obvious, but 84 is indeed divisible by 7.

By continuing this process, we find that there aren't any more whole number pairs that multiply to 84. Therefore, the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84. These numbers, when paired appropriately, will all produce 84 when multiplied.

Method 2: Prime Factorization

A more sophisticated method involves finding the prime factorization of 84. Prime factorization means expressing a number as the product of its prime factors. Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).

To find the prime factorization of 84, we can use a factor tree:

      84
     /  \
    2   42
       /  \
      2   21
         /  \
        3   7

This factor tree shows that 84 can be expressed as 2 x 2 x 3 x 7, or 2² x 3 x 7. This prime factorization is unique to 84; every composite number has only one prime factorization.

Knowing the prime factorization allows us to find all the factors of 84. We can combine the prime factors in various ways:

• 2 x 2 x 3 x 7 = 84
• 2 x 2 x 3 = 12
• 2 x 2 x 7 = 28
• 2 x 3 x 7 = 42
• 2 x 7 = 14
• 2 x 3 = 6
• 2 x 2 = 4
• 3 x 7 = 21
• 2 = 2
• 3 = 3
• 7 = 7
• 1 = 1

This method systematically generates all the factors we found in Method 1.

Method 3: Using Division

A third method involves systematically dividing 84 by each whole number, starting from 1, to check for divisibility. If the division results in a whole number, both the divisor and the quotient are factors. This method is slightly less efficient than the prime factorization method, but it works equally well.

Explanation of the Mathematical Principles Involved

The process of finding factors and performing prime factorization is fundamentally linked to the concept of divisibility. Understanding divisibility rules (e.g., a number is divisible by 2 if it's even; a number is divisible by 3 if the sum of its digits is divisible by 3) greatly simplifies the process.

Prime factorization is a cornerstone of number theory. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers (disregarding the order of the factors). This theorem underpins many important concepts in mathematics, including cryptography and the study of algebraic structures.

Frequently Asked Questions (FAQ)

• Q: What are the factor pairs of 84?

A: The factor pairs of 84 are (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), and (7, 12).

• Q: How many factors does 84 have?

A: 84 has 12 factors.

• Q: Is 84 a perfect number?

A: No, 84 is not a perfect number. A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). The sum of the proper divisors of 84 (1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28) is 108, which is not equal to 84.

• Q: How can I find the factors of other numbers?

A: You can use any of the methods described above: systematic listing, prime factorization, or repeated division. Prime factorization is generally the most efficient method for larger numbers.

• Q: What is the significance of prime factorization?

A: Prime factorization is fundamental in number theory and has applications in cryptography and other areas of mathematics. It provides a unique representation of a number and helps in understanding the divisibility properties of that number.

Conclusion: Beyond the Numbers

Finding the numbers that multiply to 84 is more than just a simple multiplication problem. It provides a practical application of fundamental mathematical concepts like factors, multiples, prime factorization, and divisibility. Mastering these concepts lays a solid foundation for more advanced mathematical studies. The systematic approaches outlined above – systematic listing, prime factorization, and division – offer versatile tools for tackling similar problems involving the factors of any number. By understanding these methods, you'll not only be able to find the factors of 84 but also develop a deeper appreciation for the elegance and structure inherent in number theory. Remember, the key is practice and exploring different approaches to solidify your understanding.