Why exactly 30 days to review the laws of the upcoming holidays? Is there anything about the number 30 that makes it significant?
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We find by the laws of mentioning rain and asking for rain that one who is on doubt if he did or did not do so, if it is after 30 days since the change need not go back. Some hold that a correct habit can be assumed based on how many times one has said it over 30 days (at least 90 times) and others hold it is based on 30 days of practice. According to the latter opinion this can be why we start learning up the laws of a holiday 30 days before; to habituate ourselves with thinking about those laws so that it comes more natural by the time the holiday actually hits.
The Gemara (Pesachim 6a-b) derives the 30-day period from the fact that at the time of the first Pesach in the desert (in the year after the Exodus), Moshe told the people the laws of Pesach Sheni - which occurs one month later.
As for the reasoning behind this: R. Schneur Zalman of Liadi (in his Shulchan Aruch, Orach Chaim 429:1) summarizes Beis Yosef and others, who point out (based on Avodah Zarah 5b) that 30 days are needed to find good unblemished animals for the sacrifices one must bring for each Yom Tov (korban pesach, olas re'iyah, etc.). So even after the Beis Hamikdash was destroyed, this rule remained in effect.