The Fundamental Theorem of Arithmetic

1. Understanding the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic tells us that every composite number can be broken down into a product of prime numbers. You have already learnt how to factorise numbers using a factor tree. Now we go one step deeper to understand what this really means.

In simple words: every composite number is made by multiplying prime numbers, and this way of breaking it is always unique (the order may change, but the primes themselves stay the same).

2. Building Numbers Using Primes

Think of prime numbers as the basic building blocks of all natural numbers. Just like all buildings are made from basic materials, all composite numbers are made from primes. If you take some primes and multiply them in any way you like, you will keep creating different positive integers.

2.1. Making Numbers From a Set of Primes

Suppose you take a list of primes: 2, 3, 7, 11, and 23. By multiplying these primes in different combinations and allowing repetition, you can create many numbers:

\(7 \times 11 \times 23 = 1771\)
\(2 \times 3 \times 7 \times 11 \times 23 = 10626\)
\(2^2 \times 3 \times 7 \times 11 \times 23 = 21252\)
\(3 \times 7 \times 11 \times 23 = 5313\)
\(2^3 \times 3 \times 7^3 = 8232\)

There is no end to how many numbers we can create this way. So even with a few primes, we get an infinitely large list of composite numbers.

2.2. What Happens If We Use All the Primes?

You may wonder: if we consider all prime numbers (and there are infinitely many), can we generate every composite number? This is the big question the theorem aims to answer. To explore this, we look at how numbers are factorised.

3. Factorising a Number Using a Factor Tree

To understand whether every composite number can be made from primes, let us use a factor tree, something you already know. A factor tree breaks a number into smaller factors until all remaining factors are prime.

Let us take a large number, say 32760, and break it down:

3.1. Factor Tree of 32760

The tree keeps dividing the number by smaller primes:

32760 ÷ 2 = 16380
16380 ÷ 2 = 8190
8190 ÷ 2 = 4095
4095 ÷ 3 = 1365
1365 ÷ 3 = 455
455 ÷ 5 = 91
91 ÷ 7 = 13

All the final numbers — 2, 3, 5, 7, 13 — are prime numbers. So the prime factorisation of 32760 is:

\(32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13\)

4. What the Theorem Says

From the factor tree, we notice something: once a number is completely broken into primes, the same primes always appear. Maybe their order changes, but the list of primes never changes. This brings us to the important theorem.

4.1. Formal Statement

Theorem: Every composite number can be expressed as a product of prime numbers, and this factorisation is unique, except for the order in which the prime factors occur.

4.2. What 'Unique' Really Means

Suppose you factorise 32760 again using a different method. You may divide by 3 first, or 5 first, or use a totally different path. But when you finish completely, you will still end up with the same set of prime numbers: three 2’s, two 3’s, one 5, one 7, and one 13.

So even if the order changes, the primes themselves do not change.

4.3. General Form of Prime Factorisation

If a composite number \( x \) is written as:

\(x = p_1 p_2 p_3 \dots p_n\)

where \( p_1 \leq p_2 \leq p_3 \leq \dots \leq p_n \) are primes in ascending order, then this factorisation is unique.

When we combine repeated primes, we get powers of primes, like: