Think of numbers as ingredients in a recipe. Some numbers don't have ingredients, like sugar, an apple, salt, butter. They're complete in themselves. These are prime numbers.
Apple pie, sponge cake, bread, these contain ingredients. Numbers like 38 and 60 have ingredients. 38 has two: 19 and 2; 60 has four: 2, 2, 3, 5. 19 and 60 have an ingredient in common: 2.
If we lay the ingredients out on a table: 2, 2, 3, 5, 19 we can see we only have to put two 2's, not three, because one is shared by 38 and 60.
If we pick up 2 and 19 we can bake 38; if we pick up both 2's, 3 and 5 we can bake 60.
These ingredients are called factors. We bake them, combine them, by multiplying them.
If we combine the ones we laid out on the table we get 2*2*3*5*19=1140. So 1140 has all the ingredients we need to bake 38 or 60, and that makes it the smallest number that 38 and 60 both divide into.