Let’s assume we are talking about whole numbers (integers), so that we can distinguish between prime numbers and composite numbers.
A common factor can be separated out using the distributive property.
So let’s take an example where x is a common factor: x(a+b)=ax+bx.
But a+b is also a quantity and a factor of ax+bx. a+b can be represented by a single quantity q, that is, q=a+b. So the product is xq, a composite number.
Since all composite numbers can be represented as the product of two or more other numbers, x(a+b), (x+y)(a+b), etc., including polynomials as factors, all factorisations share the same property: they are all the product of at least two factors.