On the Day Allah declared His Deen perfected (Q5v3), Prophet Muhammad [saws] said:
"The pattern according to which time is reckoned is always the same.  "With Allah, the months are twelve in number.  "Four of them are holy.  "Three of these are successive and one (Rajab)  occurs singly, between the months of Jumada and Shaban." ( as reported by Abu Bakr [ra] in Sahih Bukhari Vols. 5, 6, & 9; and same in Sunan Abu Dawud )

Prophet Muhammad [saws] warned his followers against following the People of the Book into the hole of a lizard [as reported by Abu Sa'id al-Khudri [ra], in Bukhari, Vol. 4 #3456 (The Book: of the Stories of the Prophets ) and Vol. 9 #7320 (The Book: Holding Fast to the Qur'an and the Sunnah) ].

Musa [saws] Obeyed (click here to download PDF)
When Moses [saws] lead his people out of slavery from Egypt, Allah [swt] commanded him:  " Lead your people out of darknesses into the Light, and remind them of the days of Allah [Q14v5].  Moses [saws] was one who obeyed.  He taught bani Israel the proper reckoning for the days of the week, and the months of the year:

Jewish Days
The Jewish day begins at sunset. The status of the period between sunset (the disappearance of the sun behind the horizon) and nightfall (the emergence of three medium-sized stars) is doubtful. For some purposes, it is treated as part of the previous day, e.g. at the end of Shabbat, when the prohibition of creative activities (melacha) remains in force until nightfall. With the exception of the Shabbat, the weekdays have no names. They are simply numbered:

1. yom rishon (First-day)
2. yom sheni (Second-day)
3. yom sh'lishi (Third-day)
4. yom revi'i (Fourth-day)
5. yom chamishi (Fifth-day)
6. yom shishi (Sixth-day)

The week culminates in the seventh day, the Holy Shabbat

Jewish Months
The Jewish month is based on the lunar or synodic month, the time it takes for the moon to circle the earth. Since the exact duration of one revolution is a little over 29.5 days, the length of the months normally alternates between 29 and 30 days. A month of 30 days is called male ('full'), one of 29 days chaser ('defective'). There are two months which are male in some years and chaser in others.

The month begins with the appearance of the new moon. In the time of the Temple, the Sanhedrin (the highest court) sanctified the new month when two witnesses had actually sighted the moon. In the middle of the fourth century C.E., a fixed calendar was introduced.

In the Torah, the months are numbered; the first is the one in which the Exodus (hijrah) from Egypt occurred (Yetziat Mitzrayim; cf. Shemot [Exodus] 12:2).

Yehud Surrendered the Deen ul Qayyim Calendar in Babylon
Nebuchanezzar defeated the Jews, and enslaved them in Babylon (now Iraq).
Subsequently, names of Babylonian origin were adopted for the months of the Jewish calendar:

1. Nisan (30 days)
2. Iyyar (29 days)
3. Sivan (30 days)
4. Tammuz (29 days)
5. Av (30 days)
6. Elul (29 days)
7. Tishri (30 days)
8. Cheshvan (29 or 30 days)
9. Kislev (30 or 29 days)
10. Tevet (29 days)
11. Sh'vat (30 days)
12. Adar (29 days)

Please note, that the first month of the Jewish calendar began with Bani Israel's hijra and the names of the months corresponded with their number.  They adopted the present names after their sojourn in Babylon, and adopted fixed months based on the solar cycle during the fourth Christian century.

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