Understanding prime factors method of finding a LCM I am able to understand why the LCM of prime numbers is just their product; but for non primes we get their prime factors and choose the 'maximum occurrence' of every prime factor and multiply them, I lose my understanding here, why choose the 'maximum occurrences' ?
 A: In short, this relies on unique prime factorization. Every number can be written uniquely as a product of primes; and if a prime occurs a certain number of times in each number, it must occur at least as often in their LCM.
Let's look at this in terms of an example. Consider $2^3 \cdot 3 \cdot 5 = 120$ and $2^2 \cdot 3^3 \cdot 7 = 756$. Each of these numbers will divide their LCM (really, the LCM of two numbers $a$ and $b$ is the smallest number that is divisible by both $a$ and $b$). So in particular, it will divisible by each of the prime-powers that divide these numbers. $2^3$ divides $120$, so $2^3$ will divide the LCM of $120$ and $756$. So there will be at least $3$ 'occurrences' (to use your phrase) of the prime $2$ in the LCM. There won't be a need for any more, because only two occurrences of $2$ occur in $756$. Having $2^3$ thus satisfies both numbers' divisibility needs.
Similarly, if $120$ divides a number, then in particular it is divisible by $3$. But in fact we need three powers of $3$ from $756$. '
This argument carries on, to numbers besides $120$ and $756$. You need the 'maximum occurrence' of each prime factor because the LCM needs to be a multiple of that maximum prime factor.
