>>12198777Prime numbers are (in a sense) the building blocks of all natural numbers. The Fundamental Theorem of Arithmetic (FTA) says that every natural number has a unique prime factorization, e.g. 6 = 2 x 3 = 3 x 2 (which is just the same factors re-arranged, so it doesn't count as a different factorization), so you can never get 6 by choosing any distinct factor expression, say 2 x 2, 3 x 3, 2 x 5, etc.
Once you know this, you can apply the idea to compare the factors in any two natural numbers. The biggest thing that you can factor out between two such numbers are "the most" prime factors that you can factor out-literally, algebraically pull out which are common to both.
Say you're comparing 54 and 66. You can start intuitively. They're both even, so we can factor out 2: 27 and 33, now. Okay, we can get a 3 out, just to look at it: now we're down to 9 and 11. Hm, doesn't seem we can go any further. The reason why we can't go any further is because these two numbers are coprime, or relatively prime (same thing). This is the situation where two numbers have a GCD of 1, i.e. no prime factors in common at all. Notice how the idea of simplifying fractions is related to this prime number business.
Let's do the same thing again but in a more formal way. Let's re-write 54 and 66 as their prime factors. This works out to
2 x 3^3 ; 2 x 3 x 11.
We can pull out a 2, and a 3. And those are all the prime terms common to both numbers. So the GCD of 54 and 66 is 6, and that's why. If you know the first few prime numbers cold, you can cook up examples mentally for smaller (certain three and four digit) numbers.
If you set them up as a fraction, you can CANCEL the 2x3 term(s) (divided by itself), because it's equal to 1. This is how reducing fractions works.
Fun fact: the LCM procedure is much the same thing, flipped around.
