Factorise 84 By Division Method

Factorising 84 by the Division Method: A Comprehensive Guide

Factorisation is a fundamental concept in mathematics, crucial for simplifying expressions, solving equations, and understanding number properties. This article provides a detailed explanation of how to factorise the number 84 using the division method, a process that goes beyond simply finding the factors and delves into the underlying principles of prime factorisation. We will explore the steps involved, the rationale behind each step, and address common questions. Learning to factorise efficiently is a valuable skill that builds a strong foundation for more advanced mathematical concepts.

Understanding Factorisation and Prime Numbers

Before diving into the division method, let's clarify some key terms. Factorisation is the process of breaking down a number into smaller numbers that, when multiplied together, give the original number. These smaller numbers are called factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because 1 x 12 = 12, 2 x 6 = 12, and 3 x 4 = 12.

A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and so on. Prime numbers are the building blocks of all other whole numbers. The process of finding the prime factors of a number is known as prime factorisation. The division method is a common technique for prime factorisation.

Step-by-Step Factorisation of 84 by Division

Now let's factorise 84 using the division method. This method involves repeatedly dividing the number by its smallest prime factor until we are left with 1.

Step 1: Start with the smallest prime number.

The smallest prime number is 2. We check if 84 is divisible by 2. Since 84 is an even number (ends in 4), it is divisible by 2.

84 ÷ 2 = 42

Step 2: Continue dividing by the smallest prime factor.

Now we have 42. Again, 42 is an even number, so it's divisible by 2.

42 ÷ 2 = 21

Step 3: Find the next smallest prime factor.

21 is no longer divisible by 2. The next smallest prime number is 3. Is 21 divisible by 3? Yes, because 2 + 1 = 3, and 3 is divisible by 3.

21 ÷ 3 = 7

Step 4: The quotient is a prime number.

We are left with 7, which is a prime number. This means we have reached the end of the prime factorisation process.

Step 5: Write the prime factorisation.

We can write the prime factorisation of 84 as 2 x 2 x 3 x 7, or more concisely as 2² x 3 x 7. This means that 2 multiplied by itself twice, multiplied by 3, and multiplied by 7, equals 84.

Visual Representation: Factor Tree

A factor tree is a helpful visual aid for understanding prime factorisation. Here's how a factor tree would look for 84:

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

The factor tree shows the branching process of dividing by prime factors until we reach only prime numbers at the bottom. Following the branches from the top down and multiplying the numbers at the end gives us the prime factorisation: 2 x 2 x 3 x 7 = 84.

Why This Method Works: Understanding the Fundamental Theorem of Arithmetic

The division method relies on the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be represented uniquely as a product of prime numbers (ignoring the order of the factors). This theorem guarantees that our prime factorisation of 84 (2² x 3 x 7) is the only possible prime factorisation, excluding variations in the order of the factors. This uniqueness is crucial in many areas of mathematics.

Dealing with Larger Numbers: A Systematic Approach

The division method is efficient even for larger numbers. Let's consider a slightly more challenging example: factorising 360.

1. Start with 2: 360 ÷ 2 = 180
2. Continue with 2: 180 ÷ 2 = 90
3. Continue with 2: 90 ÷ 2 = 45
4. Move to 3: 45 ÷ 3 = 15
5. Continue with 3: 15 ÷ 3 = 5
6. End with 5 (a prime number): 5 ÷ 5 = 1

Therefore, the prime factorisation of 360 is 2 x 2 x 2 x 3 x 3 x 5 = 2³ x 3² x 5.

The key is to systematically test prime numbers, starting with the smallest, and continue until you reach a quotient of 1.

Applications of Factorisation

Factorisation isn't just an abstract mathematical exercise; it has numerous practical applications:

• Simplifying Fractions: Factorisation helps simplify fractions by identifying common factors in the numerator and denominator. For example, simplifying 84/126 requires finding the prime factorisation of both numbers. 84 = 2² x 3 x 7 and 126 = 2 x 3² x 7. The common factors are 2, 3, and 7. Simplifying, we get 2/3.

• Solving Algebraic Equations: Factorisation is essential for solving quadratic and higher-degree equations. The ability to factor expressions like x² + 5x + 6 = (x+2)(x+3) is crucial for finding the roots of the equation.

• Cryptography: Prime factorisation plays a vital role in cryptography, particularly in public-key cryptography systems like RSA. The security of these systems relies on the difficulty of factoring very large numbers into their prime factors.

• Number Theory: Prime factorisation is a fundamental concept in number theory, a branch of mathematics that studies the properties of numbers.

Frequently Asked Questions (FAQ)

• Q: What if I miss a prime factor?

  • A: If you miss a prime factor during the division method, you won't get the complete prime factorisation. You will end up with a composite number (a number that is not prime) instead of 1. Always systematically check for divisibility by each prime number in ascending order.

• Q: Can I use any order of prime factors?

  • A: While the order of factors doesn't change the result (commutative property of multiplication), it's generally best practice to use prime factors in ascending order for consistency and ease of understanding.

• Q: What if the number is already a prime number?

  • A: If the number is already prime (like 7, 11, 13, etc.), its prime factorisation is simply the number itself.

• Q: Are there other methods for factorisation?

  • A: Yes, there are other methods like the factor tree method (which we visually represented above), and using divisibility rules to quickly identify factors. However, the division method provides a systematic and reliable approach, especially for larger numbers.

Conclusion

The division method provides a straightforward and efficient approach to factorising numbers, leading to their prime factorisation. Understanding this method not only improves your arithmetic skills but also builds a solid foundation for more advanced mathematical concepts. The process involves systematically dividing by prime numbers until you reach 1, resulting in a unique prime factorisation. Remember, this process is based on the Fundamental Theorem of Arithmetic, a cornerstone of number theory. Mastering factorisation, therefore, opens doors to a deeper appreciation of number properties and their numerous applications in various mathematical fields and beyond. Practice regularly, and you'll find factorisation becoming increasingly intuitive and efficient.