2. The fundamental theorem of arithmetic

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

 

Explanation:

 

Let us take the composite number \(N\).

 

Decompose the number \(N\) into the product of primes.

 

 

Here, the number \(N = x_1 \times x_2\). But, both \(x_1\) and \(x_2\) are again a composite number. So, factorise it further to obtain a prime number.

 

The prime factors of \(x_1 = p_1 \times p_2\).

 

The prime factors of \(x_2 = p_3 \times p_4\).

 

We get, \(N = p_1 \times p_2 \times p_3 \times p_4\) where \(p_1\), \(p_2\), \(p_3\) and \(p_4\) are all prime numbers.

 

If we have repeated primes in a product, then we can write it as powers.

 

 

In general, given a composite number \(N\), we factorise it uniquely in the form $N={p_1}^{q_1}\times {p_2}^{q_2}\times {p_3}^{q_3}\times ...\times {p_n}^{q_n}$ where $p_1,p_2,p_3,...p_n$ are prime numbers, and $q_1,q_2,q_3,...q_n$ are natural numbers.

 

Thus, every composite number can be expressed as a product of primes apart from the order.