Divisors & Multiples
A divisor divides another number evenly, and a multiple is that number taken a whole number of times. The divisors of 12 are 1, 2, 3, 4, 6, and 12, while the multiples of 3 run on as 3, 6, 9, and beyond. The greatest common divisor and the least common multiple are what you use to share things equally or match repeating cycles. Slide the two numbers and the Venn diagram and the number line show both values.
Splitting something into equal groups and stacking the same gap over and over do not use the same list. The numbers that divide evenly never grow past the number itself, so that list ends. The numbers you get by taking times keep going at the same gap. When you look at two numbers together, you pick the largest shared divisor and the smallest shared multiple as two different jobs, so the slot that shares evenly and the slot that comes back together do not get mixed. If you memorize only one list, you already do not know which one to reach for.
The two sliders pick from 6 to 48, and the first picture has a blue circle overlapping a gold one. Numbers only in the left circle are divisors of the first number, numbers only in the right circle are divisors of the second, and the common divisors sit in green down the middle overlap. At the bottom the largest of those overlapping numbers is written as the GCD. The second picture has two horizontal lines, and on each line the multiples of that number are marked as dots, up to 8 of them. Dots that have the same value on both lines are larger and green, and a dashed line drops from the leftmost of those values, with the LCM written there. The dots and circles stay where they are drawn; they do not travel along the line. The fact that the product of the two numbers equals the product of the GCD and the LCM is not written on this picture.
Common divisors and common multiples both use the word common, so it is easy to write the large one as the LCM or the small one as the GCD. A divisor list ends inside the number, so you take the largest overlapping value. A multiple list keeps going larger, so you take the smallest overlapping value. The directions are opposite, and even with the same pair the value you pick depends on which picture's overlap you are looking at. The practice items can wait until you know which picture you are reading. Decide first whether you are choosing from the circles or from the lines, and the two names will not swap.