There is no biblical reference to 613 commandments, although the later rabbinic leaders claimed that all 613 commandments are alluded to within the Ten Commandments.
The first actual reference to 613 commandments is found in a lengthy Talmudic passage. There, Rabbi Simlai (third century A.D.) is quoted as saying, “Six hundred and thirteen precepts were communicated to Moses, three hundred and sixty-five negative precepts, …and two hundred and forty-eight positive precepts…” (b. Makkot 23b-24a; we will return to this passage at the end of this answer). Based on this comment, medieval Jewish scholars, sought to come to agreement as to the exact enumeration and delineation of the 613 commandments, since there is a good deal of ambiguity in counting.
For example, would you count “Be fruitful and multiply,” spoken in Genesis 1:28 and repeated in Genesis 9:1, to be one of the 613 commandments? The rabbis did, recognizing it as the first word of command given to the human race. What about commandments that overlap each other? Should these be subdivided?
Questions such as these caused these scholars to come up with slightly different counts-there was essential agreement about the vast majority of these commandments-and the most famous compilation of all, Maimonides’ Book of the Commandments, actually counted only 606 explicit Torah commandments, the last seven consisting of rabbinic commandments, namely:
(1) The command to wash the hands before eating (note the conflict over this already in Matthew 15 and Mark 7);
(2) laws regarding the Eruv, a rabbinic concept that removed some Sabbath restrictions in very specific ways;
(3) reciting a blessing before eating or partaking of any kind of pleasure;
(4) lighting the candles on the Sabbath;
(5) celebrating the holiday of Purim, which is referred to in the book of Esther but whose observance is not mandated there (see Esther 9:23-32);
(6) celebrating Hanukkah, a holiday commemorating events that took place in the second century B.C., well after the Tanakh was already completed;
(7) reciting the Hallel prayer on certain occasions.
Although some Jewish leaders differed with Maimonides in including these rabbinic commandments in the count of 613, all traditional Jews recognize these seven commandments as divinely given through the authority of the rabbis. Thus, as noted above (#14) when a Jewish woman lights the Sabbath candles, she blesses God “who commanded us to kindle the Sabbath candles,” even though this is not written in the Torah but rather comes from the rabbis.
The Hebrew acronym for 613 is taryag, as the Hebrew letter t equals 400, r equals 200, y equals 10, and g equals 3. Thus the 613 commandments are known in Hebrew as taryag mitzvoth. In keeping with Rabbi Simlai’s statement, the commandments are divided into negative and positive precepts (that is, “Thou shalt not” and “Thou shalt”). Because of the destruction of the Temple and the lack of a functioning Temple priesthood, Jews today can observe, as written in the Torah, only 77 out of the 365 negative commandments and 194 out of 248 positive commandments. This is one of the major reasons that religious Jews pray three times daily for the rebuilding of the Temple and redemption of their people: They are eager to perform the rest of the commandments!
For a fascinating passage in the Talmud that states that Habakkuk ultimately reduced the 613 commandments to one – namely, Habakkuk 2:4, the just will live by faith – (see Talmud, Makkoth 24a, and more on this question in the book 60 Questions Christians Ask About Jewish Beliefs and Practices).
Yes, the concluding comment is that the 613 commandments were reduced to one, as expressed in Habakkuk 2:4, “The just shall live by faith”-a favorite text of Paul! (See Romans 1:17; Galatians 3:11.)
Of course, this is not to say that the Talmudic rabbis were Christians, but it is to say that when Paul boiled everything down to the just living by faith, he was in good company.