To make sure we are starting on the same page... The Documentary Hypothesis answers questions that began piling up about the Law of Moses or Books of Moses referring to the first five books of the Bible (Genesis,Exodus,Leviticus,Numbers,Deuteronomy). Questions like:
How did Moses describe his own death?
Why are there two creation stories?
Why are there two sets of animals in the ark?
Why are there two sets of ten commandments?
Why do each of these duplicate stories refer to God in a different name (Elohim or Yahweh for example. Compare how Genesis 6 refers to two of every sort of animal using the term "God" in our modern translation and Genesis 7 refers to seven pairs of animals using the word "Lord" in our modern translation).
Any one of these could easily be explained by themselves but it became suggested that the reason for the confusion was because a later redactor combined the four sources (JEPD) into one. We also know that these sources were collected and combined at a later date. It seems that Deuteronomy and Leviticus, for example, were written over many centuries. The closest I could come to early Christians being aware of this are the Jewish Christians who followed James the Just. The Nasaraeans (which some scholars believe were Jewish Christians) apparently understood this long before modern scholars:
“The Nasaraeans. They were Jews by nationality... They practiced Judaism in all respects, and scarcely had any beliefs beyond those of the Jewish sects I have mentioned. They too had acquired circumcision, and they kept the same Sabbath and were attached to the same feasts, but they did not introduce fate or astrology. They too recognized the fathers in the Pentateuch from Adam to Moses, who had been conspicuous for excellence of piety... But they would not accept the Pentateuch itself. They acknowledged Moses and believed that he had received legislation – not this legislation, however, but some other [see Jeremiah 8:8]. And so, though they were Jews who kept all the Jewish observances, they did not offer sacrifice or eat meat. They considered it unlawful to eat meat or make sacrifices with it. They claimed that these books are fictions and that none of these customs were instituted by the fathers”
Epiphanius, Panarion, Against Nasaraeans
Jesus refers to Moses frequently in the gospels:
"He said to them, “Because of your hardness of heart Moses allowed you to divorce your wives, but from the beginning it was not so". Matthew 19:8
For Moses said, ‘Honor your father and mother,’ and, ‘Anyone who curses their father or mother is to be put to death.’ Mark 7:10
"Have you not read in the Book of Moses, in the account of the burning bush, how God said to him, ‘I am the God of Abraham, the God of Isaac, and the God of Jacob’ Mark 12:26
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There are so many permutations of possibilities that I don't know where to start. The question becomes, if some of the earliest Christians knew about the problem with the books of Moses, did Jesus also know about this? Did Jesus use hyper-legalism and simply work within the confines of the law of his day? Or did these laws come from God? Did Jesus simply refer to the "laws of Moses" or "Moses said" in a figurative way since that is what they were known as at that time? Or did Jesus take the law 100%? I am not sure the best way to reconcile this.