Prime Factorization Calculator

Understanding Prime Factorization

What is Prime Factorization?

Prime factorization is the process of breaking down a number into the product of its prime factors. A prime factor is a prime number that divides the original number without leaving a remainder.

The Concept

Every positive integer greater than 1 can be represented uniquely as a product of prime numbers. This is known as the Fundamental Theorem of Arithmetic.

The Process

To find the prime factorization of a number:

1. Start with the number \(n\)
2. Find the smallest prime number \(p\) that divides \(n\) evenly
3. Divide \(n\) by \(p\)
4. Repeat steps 2 and 3 with the quotient until the quotient itself is a prime number

Example

Let's find the prime factorization of 84:

\[ 84 = 2 \times 42 \] \[ 42 = 2 \times 21 \] \[ 21 = 3 \times 7 \]

Therefore, \(84 = 2 \times 2 \times 3 \times 7\) or \(2^2 \times 3 \times 7\)

Visual Representation

This tree diagram shows the prime factorization of 84.