Asara B'Tevet / עשרה בטבת

Tenth of Tevet (Hebrew: עשרה בטבת, Asarah BeTevet), the tenth day of the Hebrew month of Tevet, is a minor fast day in Judaism. It is a "low fast" observed from sunrise to sunset. The day has no relationship to Hanukkah, but it happens to follow that festival by a week. Whether the 10th of Tevet falls 7 or 8 days after Hanukkah depends on whether the preceding month of Kislev has 29 or 30 days in the relevant year.
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List of Dates

Asara B'Tevet begins in the Diaspora on:

Torah Portion: Exodus 32:11 - 34:10

1: Exodus 32:11-14 (4 p'sukim)
2: Exodus 34:1-3 (3 p'sukim)
3: Exodus 34:4-10 (7 p'sukim)

References

The Jewish Holidays: A Guide & Commentary
Rabbi Michael Strassfeld
Hebrew - English Bible
Mechon Mamre
Tanakh: The Holy Scriptures
Jewish Publication Society
"Tenth of Tevet" in Wikipedia: The Free Encyclopedia
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