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# Fundamental Theorem of Arithmetic

October 31, 2023 written by Rida Mirza

Factorization is the process of expressing a number as the product of its factors. The Fundamental Theorem of Arithmetic describes the unique prime factorization of integers. This theorem is a key concept in number theory and has several important applications in mathematics.

## Prime and Composite Numbers

A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself.

For example, 2, 3, 5, 7, 11 are prime numbers.

A composite number is an integer greater than 1 that is not prime. It has at least one positive divisor other than 1 and itself.

For example, 4, 6, 8, 9, 10 are composite numbers.

## Prime Factorization

Every composite number can be expressed as the product of prime factors. This is known as the prime factorization of the number.

### Example

12 = 2 x 2 x 3

Here 2, 2, 3 are prime factors of 12.

Prime factorization can be used to find the greatest common divisor (GCD) and lowest common multiple (LCM) of two numbers easily.

## Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented as the product of prime numbers. Moreover, this representation is unique, apart from the order of the factors.

From this theorem we also see that not only a composite number can be factorized as the product of their primes but also for each composite number the factorization is unique.

In simple words, there exists only a single way to represent a natural number by the product of prime factors. This fact can also be stated as:

The prime factorization of any natural number is said to be unique for except the order of their factors.

In general, a composite number “a” can be expressed as,

a = p1 p2 p………… pn, where p1, p2, p………… pn are the prime factors of a written in ascending order i.e. p1≤p2≤p………… ≤pn.

## Example

Consider the number 36. By using the Fundamental Theorem of Arithmetic, we can find its prime factorization:

• We divide 36 by 2, which gives us 2 x 18.
• 18 can be divided by 2, resulting in 2 x 9.
• 9 can’t be divided by 2, but it’s a perfect square of 3 x 3.

Now, we have the prime factorization of 36: 2 x 2 x 3 x 3. This is the unique way to express 36 as a product of prime numbers.

## FAQs

### What is a factor?

A factor is a number that divides evenly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, 12.

### What is the difference between prime and composite numbers?

Prime numbers have exactly two factors – 1 and the number itself. Composite numbers have more than two factors.

### What are some examples of prime numbers?

2, 3, 5, 7, 11, 13, 17, 19 are prime numbers. Other examples include 37, 61, 97, 127 etc.

### How do you find the prime factorization of a number?

Keep dividing the number by prime numbers successively until you reach the prime factors. For example, the prime factorization of 60 is 2 x 2 x 3 x 5.

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