What is the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic says that every whole number greater than one is either a prime number, or the product of two or more prime numbers. No matter how, or in what order, you break the number down into its factors you will end up with exactly the same prime factors.

For example, 6 is 2 x 3 (or 3 x 2). There’s no other way to make a 6 out of prime numbers except out of a 2 and a 3. It doesn’t matter what order you put the factors, just what the factors are. Keep in mind that this is all about multiplication — not addition.

As another example, 28 is 2 x 2 x 7. You can rearrange the order, but if you take away, add, or change any of the prime numbers, you will end up with a different number. This theorem is so important that an entire branch of mathematics is based on it: Algebraic Number Theory. However, Algebraic Number Theory takes it in a surprising direction by investigating interesting kinds of numbers (not the whole numbers that we are talking about here) in which the Fundamental Theorem of Arithmetic may or may not be true.

Prime Numbers are like building blocks.

You can think of prime numbers as being building blocks from which all numbers are made (whole numbers greater than one, that is). Another way to think of it is that the prime factors make a recipe for a number, and that there is only one recipe for each number.

Remember that the Fundamental Theorem of Arithmetic is about multiplication. Occasionally students forget and get a bit confused, but there is nothing like the Fundamental Theorem of Arithmetic for addition. Another possible source of confusion is when factors are found that are themselves composite numbers. For example, 12 is 4 x 3, but it’s also 6 x 2. So is 12 an exception to the Fundamental Theorem of Arithmetic? No, of course not. 4 is 2 x 2, and 6 is 2 x 3, so:

When we say 12 is 4 x 3, that means 2 x 2  x3. When we say 12 is 6 x 2, that means 3 x 2 x 2. In both cases, 12 is made of two 2s and one 3. Since when you factor 12 you always get two 2s and one 3, we say that it is the unique factorization of 12.

We use prime factorization to break a number into its constituent prime factors.

Next Topic: Prime Factorization