First, Find the Prime Factorization of 84: A Step-by-Step Guide

Understanding prime factorization is essential for mastering number theory and solving a wide range of math problems, from simplifying fractions to finding least common multiples. If you’re wondering “First, find the prime factorization of 84,” you’ve come to the right place. In this article, we’ll walk you through the complete process of breaking down the number 84 into its prime factors step by step.

Understanding the Context

What Is Prime Factorization?

Prime factorization is the process of determining which prime numbers multiply together to equal a given integer. Every composite number can be expressed uniquely as a product of prime factors—this is guaranteed by the Fundamental Theorem of Arithmetic.

Step-by-Step: Factorizing 84

First, find the prime factorization of 84:

Key Insights

Let’s find the prime factors of 84 by following these clear, logical steps.

Step 1: Start with the smallest prime number, 2

Since 84 is an even number, it’s divisible by 2.

84 ÷ 2 = 42
Now we have: 84 = 2 × 42

Step 2: Factor 42 next

42 is also even, so again divisible by 2:

42 ÷ 2 = 21
Now: 84 = 2 × 2 × 21

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Set \(k = 0\): \(1 = A(2) \Rightarrow A = \frac{1}{2}\)
Set \(k = -2\): \(1 = B(-2) \Rightarrow B = -\frac{1}{2}\)
\frac{1}{k(k+2)} = \frac{1/2}{k} - \frac{1/2}{k+2} = \frac{1}{2} \left( \frac{1}{k} - \frac{1}{k+2} \right)

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Final Thoughts

Step 3: Factor 21

21 is odd but divisible by 3:

21 ÷ 3 = 7
Now: 84 = 2 × 2 × 3 × 7

Step 4: Check if 7 is prime

7 is a prime number (it has no divisors other than 1 and 7), so we stop here.

Final Result: Prime Factorization of 84

Putting it all together, the prime factorization of 84 is:

84 = 2² × 3 × 7

Why Prime Factorization of 84 Matters

Breaking down 84 into primes—2² × 3 × 7—is foundational for: