Fundamental theorem of arithmetic facts for kids

The Fundamental Theorem of Arithmetic is a super important idea in number theory. It's also called the unique factorization theorem. This theorem tells us that every whole number bigger than 1 can be written as a product of prime numbers.

Think of it like this: prime numbers are the basic building blocks for all other whole numbers. For example:

• 6 = 2 × 3
• 12 = 2 × 2 × 3
• 30 = 2 × 3 × 5

The amazing part is that there's only one way to break down a number into its prime factors. Even if you find the primes in a different order, it's still the same set of primes. For instance, 12 can be 2 × 2 × 3 or 3 × 2 × 2, but it always uses two 2s and one 3.

Finding these prime numbers is called factorization. This theorem is very useful, even in things like cryptography, which is about keeping information secret and secure.

Who Figured This Out?

The first person to show that every number can be broken down into primes was Euclid, a famous Greek mathematician. Later, a detailed and correct proof was written by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae in 1801.

Why Is It True?

The proof of this theorem has two main parts. First, we show that every number can indeed be written as a product of primes. Second, we show that this way of writing it is unique, meaning there's only one set of prime factors for each number.

Every Number Can Be Factored

Let's imagine for a moment that there *are* some numbers that cannot be written as a product of primes. If that were true, there would have to be a smallest number that can't be written this way. Let's call this smallest number n.

• n can't be 1, because the theorem is for numbers greater than 1.
• n can't be a prime number, because a prime number is already a "product" of just one prime (itself!).

So, n must be a composite number, meaning it can be multiplied from two smaller whole numbers. Let's say n = a × b, where a and b are both smaller than n.

But wait! We said n was the smallest number that *couldn't* be written as a product of primes. Since a and b are smaller than n, they *must* be able to be written as products of primes.

If a is a product of primes and b is a product of primes, then n = a × b must also be a product of primes! This creates a contradiction, an impossible situation. This means our first idea (that some numbers can't be written as a product of primes) must be wrong. Therefore, every number greater than 1 *can* be written as a product of primes.

The Factors Are Unique

Now, we need to show that there's only one unique way to write a number as a product of primes.

We use a special rule called Euclid's lemma. It says: if a prime number p divides (or evenly goes into) a product of two numbers, say a × b, then p must divide a or p must divide b (or both).

Let's say we have a number that can be written in two different ways as a product of primes: P1 × P2 × P3 = Q1 × Q2 × Q3 (where P and Q are prime numbers)

Take any prime number from the first list, say P1. Since P1 divides the whole product on the left, it must also divide the whole product on the right (because they are equal).

Using Euclid's lemma, if P1 divides Q1 × Q2 × Q3, then P1 must divide Q1 or Q2 or Q3. Since Q1, Q2, and Q3 are all prime numbers, P1 must be equal to one of them.

We can then divide both sides of our equation by P1 (and the Q prime it matched). We keep doing this until all the prime factors are gone. In the end, we'll see that the list of prime factors on both sides must be exactly the same. This proves that the way we break down a number into its prime factors is truly unique!

Images for kids

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  In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem and used it to prove the law of quadratic reciprocity.

See also

In Spanish: Teorema fundamental de la aritmética para niños