How did someone discover LCM?

Question

How did someone came up with an idea that if we do prime factorization of two numbers and then multiply all the prime factors but including common ones only once, we will get a number that is the least common multiple of these two numbers?

Answer 1

Let's say you want to add up the two fractions: $$\frac{1}{2}+\frac{1}{4}$$ what would you do? you could make the denominator of both sides 8, but you wouldn't do that, naturally most people don't want the fractions to get too big so they would just multiply the numerator and denominator of the first one by $2$. you would look for what's lacking. this is something many people do before they learn what LCM even is. it's a pretty natural thing to consider.