Refresher - Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic tells us that every integer greater than 1 can be expressed as the product of prime numbers. The factorization is unique except for the order in which the primes are written. This theorem established the importance of prime numbers as the basic building block of any positive integer. All positive integers can be created from the product of primes.

A prime or prime number is an integer greater than 1 that has only 1 and itself as factors. An integer greater than 1 which is not a prime number is called composite. Other than 2, all primes are odd numbers. Every integer greater than 1 has a prime divisor.

For example 13 and 17 are primes, but 21 is composite because it is divisible by 3.
A composite number that is expressed using factors that are all prime numbers is called the prime factorization of the number.

Example: 12 = 2 • 2 • 3

Fundamental Theorem of Arithmetic

The greatest common factor (GCF) or greatest common divisor of two integers is the largest positive integer that is a common factor for both numbers.
For example: 2 is a common factor of 8 and 12, but 4 is the greatest common factor.
Two integers are relatively prime if their GCF is 1.

The least common multiple (LCM) of two non-zero integers is the smallest positive integer which is divisible by both.
For example: the LCM of 4 and 6 is 12 because it is the smallest positive integer that 4 and 6 will both divide into, or 4 | 12 and 6 | 12.
The LCM and the GCF have an interesting relationship. You can find the LCM of two integers by finding the product of the two integers then divide the product by the GCF of the two numbers. $L C M_{x, y} = \frac{x y}{G C F_{x, y}}$

This is an easy way to find the LCM of two very large numbers. Let's go back to the example for finding the GCF of 700 and 448 using the Euclidean Algorithm. The GCF is 28. To find the LCM using prime factorization would be very tedious. Using this formula we have $L C M = \frac{700 (448)}{28} = 11200$
Real numbers that are not rational are called irrational numbers. Square roots of integers that are not perfect squares are irrational.