"...whose culture has a common origin..."
In what senses is it useful to consider the societies of East Asia as having cultures with a common origin? Certainly they are now markedly different on most dimensions, but one can find lots of historical grounds for tracing influence, particularly from China outward. The details are not universally agreed upon (just who were the early pottery makers of Korea and Japan --who made the oldest pottery we know of? And what functions did Korea serve as an intermediary between China and Japan? And what will molecular genetics tell us about biological relationships?), and each of the societies has vested interests in particular versions of the narrative. Still, it's clear that China has served as a source for many borrowings by Japanese and Korean cultures, including philosophical and religious ideas, and that through much of recorded history both Korean and Japanese political leadership was involved in tribute relations with China's rulers. At the very least, China and Japan and Korea have been interacting with one another pretty much continuously (and with notable complexity) for thousands of years.