Put the prime factors into a Venn diagram and the answer appears
Prime-factorize the two numbers, then place the shared prime factors in the middle (the intersection) and the prime factors unique to each number on the two sides. Multiply the factors in the middle (the common factors) to get the GCD, and multiply all the factors to get the LCM.
Why does this work? The GCD is "the greatest of the divisors common to both numbers" — in other words, it is the product of the prime factors the two numbers have in common. The LCM is "the smallest of the multiples common to both numbers" — it is the product of all the prime factors of the two numbers, using each shared factor only once.