Unique Factorization intuitive understanding What is an easy and intuitive explanation to the unique factorisation problem:

If two prime numbers a and b are multiplied, the product (a*b) cannot be divided by a different prime number c.
I think that in this case contradiction is the most intuitive way to understand this property. So let's consider two prime numbers $p$ and $q$. As you mentioned, if you take the product $pq$ the only two positive divisors, not equal to $1$ are $p$ and $q$. 

How can you grok this intuitively? 


  It was pointed out to me that the following does not provide a necessarily intuitive rationale for why $x$ in this case must divide into $p$ or $q$. I will breakout my intuition further here to show this.
  Well, what would happen if there was a third prime, call it $x$ that divided into $y$? If $x | y \implies x | pq$. But we know that if some prime number divides into a product of two numbers it must divide either one, or the other, (or perhaps both). This is a relatively straightforward factor, and should make sense based on how division is defined!


A prime number cannot be broken down into further factors, by the definition of primes, and as such $pq$ is the unique way to write this as a product of prime factors.

If it could be written in some other manner [$a, b \neq 0$ are prime], say $(ab)q$ then we would of course have $(ab) = p$ and as such, $p$ would not be prime. If we had that $ab = pq$, then we know that $a$ must divide both sides, as must $q$ so we have that: $\frac{b}{q} = \frac{p}{a}$. But the only two factors that divide $b$ are $b$ and $1$. So then we must have that either $q = b$ or $q = 1$. Similarly, we get that $a = p$ or $a = 1$. Because we know that by definition $a, b, p, q$ are prime we must have that $a = p$ and $b = q$. As such, the factorization is unique.

If the uniqueness of the factorization is not quite enough to fully ensure that the intuition is clear, consider if we had a third prime $x$, which did divide into $pq$. We know from the above logic that $x\times c = pq$ must mean one of the following things:

$x = p \implies x$ is not a third divisor.

$x = q \implies x$ is not a third divisor.

$x = pq \implies x$ is not prime.

$x = 1 \implies x$ is not prime.

The remaining options would be that $x | p$ or $x | q$, since it cannot be that $x$ divides some combination of the factors of $p$ and $q$ since the only factors of $p$ and $q$ are one and themselves! However, for $x | p$ then we know that $x = 1$ or $x = p$ which are options above and for $x | q$ we have the same issues.

So this says that we cannot possibly have a third prime $x$ which divides $pq$ unless $x$ is in fact either $p$ or $q$.
I think that in this case contradiction is the most intuitive way to understand this property. So let's consider two prime numbers $p$ and $q$. As you mentioned, if you take the product $pq$ the only two positive divisors, not equal to $1$ are $p$ and $q$. 

How can you grok this intuitively? 


  It was pointed out to me that the following does not provide a necessarily intuitive rationale for why $x$ in this case must divide into $p$ or $q$. I will breakout my intuition further here to show this.
  Well, what would happen if there was a third prime, call it $x$ that divided into $y$? If $x | y \implies x | pq$. But we know that if some prime number divides into a product of two numbers it must divide either one, or the other, (or perhaps both). This is a relatively straightforward factor, and should make sense based on how division is defined!


A prime number cannot be broken down into further factors, by the definition of primes, and as such $pq$ is the unique way to write this as a product of prime factors.

If it could be written in some other manner [$a, b \neq 0$ are prime], say $(ab)q$ then we would of course have $(ab) = p$ and as such, $p$ would not be prime. If we had that $ab = pq$, then we know that $a$ must divide both sides, as must $q$ so we have that: $\frac{b}{q} = \frac{p}{a}$. But the only two factors that divide $b$ are $b$ and $1$. So then we must have that either $q = b$ or $q = 1$. Similarly, we get that $a = p$ or $a = 1$. Because we know that by definition $a, b, p, q$ are prime we must have that $a = p$ and $b = q$. As such, the factorization is unique.

If the uniqueness of the factorization is not quite enough to fully ensure that the intuition is clear, consider if we had a third prime $x$, which did divide into $pq$. We know from the above logic that $x\times c = pq$ must mean one of the following things:

$x = p \implies x$ is not a third divisor.

$x = q \implies x$ is not a third divisor.

$x = pq \implies x$ is not prime.

$x = 1 \implies x$ is not prime.

The remaining options would be that $x | p$ or $x | q$, since it cannot be that $x$ divides some combination of the factors of $p$ and $q$ since the only factors of $p$ and $q$ are one and themselves! However, for $x | p$ then we know that $x = 1$ or $x = p$ which are options above and for $x | q$ we have the same issues.

So this says that we cannot possibly have a third prime $x$ which divides $pq$ unless $x$ is in fact either $p$ or $q$.