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Yes. The Oral Law (Talmud) was transmitted orally by Moses to his successors from generation to generation according to the method outlined by Moses as heard from G-d. It includes rulings, laws and teachings on the Torah's verses, and existed 1500 years before Jesus. Serious students would have known it.
The Oral Law, Talmud, was written down in the 3rd Century CE (see Source below) at the direction of Judah the Prince. Previously it had been illegal to commit to writing. This law to keep the Oral Torah transmitted orally was overturned because of persecution and hardship in the exile, namely the fear that it might be lost altogether.
Here is an article by Rabbi Segal who has a table showing references in the NT that correspond to Talmud: http://rabbiarthursegal.blogspot.com/2008/01/talmudic-discourse-15.html
(That tract ate in the Talmud also mentions Saul-Paul the student of Gamliel).
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