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Isaac Moses
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Where does Baal HaTurim's commentary come from?

A lot of the Baal HaTurim's commentary on the Torah relates to how certain words or phrases have the same gematriya as other words or phrases in the Torah, or phrases that relate, in some way, to the topic at hand even if they don't appear anywhere in Tanach. This type of commentary requires not only an incredible knowledge of all of Tanach but also a mathematically-wired understanding that's not necessarily conventional. Did he have a mesorah for the gematriya-indenticalities? Was he able to look at a word or phrase and immediately come up with another word or phrase that had the same gematriya? Also, do we assume that other mefarshim such as Rashi and the Ramban knew these gematriyas or were they the Baal HaTurim's chiddushim (though "chiddush" in this context sometimes implies something that others wouldn't necessarily agree with, whereas gematriyas are indisputable)?