Split every number into a pair of factors, then keep splitting the composite one until every branch ends in a prime. Reading the leaves gives the prime factorisation — from there, HCF is the primes common to every tree (lowest power), and LCM takes every prime that appears (highest power).
Reading the answer off the trees
Every prime that shows up gets a row. HCF only keeps a prime if it appears in every number — at its lowest power. LCM keeps every prime that appears at all — at its highest power.
Where the primes overlap
Each circle is one number's prime factors. Primes sitting in every circle multiply together to make the HCF; every prime in any circle multiplies together to make the LCM.
Also called the cake / ladder method. Pick a prime that divides the numbers, divide it into every number it fits (numbers it doesn't fit are carried straight down), and repeat. For HCF only divide when a prime fits every number — stop once none do, and multiply the divisors used. For LCM keep going until every number has been reduced to 1, then multiply all the divisors down the side.
The most direct method: write out factors of each number and spot the largest one they share (HCF), or write out multiples of each number and spot the smallest one they share (LCM). Great for building intuition, clumsy for big numbers — that's exactly why the other three methods exist.
The Euclidean algorithm finds HCF without factorising at all: divide the larger number by the smaller, then replace the pair with (smaller, remainder), and repeat until the remainder is 0 — the last non-zero remainder is the HCF. For more than two numbers, chain it: HCF(a,b) first, then HCF of that result with c, and so on. (LCM is then found from HCF using the product formula.)