This idea is often called the "Fundamental Theorem of Arithmetic." Unique factorization is taught to children as young as 6th grade. When I have helped to teach number theory to high school students at PROMYS, many students take unique prime factorization for granted. Indeed, it often takes effort to convince them that the statement needs to be proved and is not just an obvious fact.
Despite the “obviousness” of unique prime factorization to the modern mind, the ancient Greeks did not know about unique prime factorization. This apparent failure of the ancient Greeks is well known to historians
of mathematics, but not known by many mathematicians. The only math text I am aware of which mentions this is Hardy and Wright's “Introduction to the Theory of Numbers.
Why did the ancient Greeks not have this result? Simply put, they lacked the words to express the statement. To a mathematician of ancient
To be sure, they groped towards this result. For example,
Apparently, the ancient Greeks, including Archimedes, Euclid and Diophantus, along with many lesser luminaries could not conceive of this idea because they lacked the language to express it. This failure is closely related to a hypothesis in psychology and linguistics known as the Sapir-Whorf hypothesis. Sapir-Whorf states that our language constrains our thought processes. There are different degrees of Sapir-Whorf. Very strong versions of the hypothesis are obviously false (if they were to be believed, translation between languages would be impossible) while very weak versions border on the trivial. This failure of the ancient Greeks is evidence for some strong form of Sapir-Whorf.
There's a lesson here for people other than psychologists. We cannot know how much we are missing simply because we lack the language to express an idea. Mathematicians over the last three centuries have taken this idea very seriously and spent much time trying to find optimal notation to express ideas. Yet, we cannot tell if there is some fundamental idea that we simply do not notice or appreciate because we lack the necessary language.