This question can be answered by establishing the approximate years when the two religions originated:
Judaism
There are at least three alternatives for the start of Judaism:
1) Abraham
Biblical tradition says that Abraham was the first in his line to worship G-d. A midrash of the common era says that Abraham realised that the idols of the gods of his father had no power and so sought the real G-d. This would place the starting date of Judaism at around 2000 BCE.
2) Moses
Judaism is also sometimes regarded as starting with Moses because God gave him the commandments, and because he is often credited with writing the first 5 books of the Bible - the Torah - which largely define Judaism. This would place the starting date of Judaism around 1400 BCE, based on the traditional date for the death of Moses.
3) Scholarly research
Based on extensive literary research, scholars now say that the first five books are composed from input from several sources, usually known as J, E, D and P. The sources known as J ("Yahwist") and E ("Elohist") seem to date back to early in the first millennium BCE. D ("Deuteronomist") dates from before 600 BCE and P ("Priestly") probably lived during the Babylonian exile. In order to establish when Judaism really began, we need to go backwards from this date to find the ealiest reliable evidence of Judaism.
During part of the tenth century BCE, the Bible says that the Hebrew people lived in a United Kingdom, ruled from the wealthy city of Jerusalem in what was to become Judah, by kings who worshipped the God of Judaism. King David conquered the well-fortified city of Jerusalem early in his reign. Because of swingeing taxes imposed by Solomon and his successor, the northern kingdom, Israel, broke away and asserted its independence. However, archaeologists tell us that there was no city of Jerusalem for David to conquer. Finkelstein goes as far as to say that the population of the whole of Judah during the relevant period was only about 40,000 - a fairly small crowd for a major football match today, and surely too small to subjugate the much larger and more prosperous northern state of Israel. Without further evidence, we can not rely on Judaism having existed during the time of Saul, David and Solomon.
We know from the Bible that the northern kingdom, Israel, was at all times polytheistic. The biblical references to the kings of Israel show every one of them as polytheistic in their beliefs. Biblical references that tell us about popular religion in Israel - what the people themselves believed - show that the nation was polytheistic from its inception until its destruction by the Assyrians.
Judaism must have begun in the southern Hebrew state of Judah. We also know from the Bible that Judah was polytheistic until the reign of Hezekia, who made a failed attempt to impose monotheism in the 7th century BCE. Arguably, if a recognisable forerunner of Judaism existed before this time, it was only a small sect, constantly at odds with the powerful kings of Judah. Hezekia's son, Manasseh, allowed polytheism to flourish once again, evidence that monotheism had not taken root among the ordinary people.
Almost a century after Hezekia, King Josiah reinstituted the reforms of his ancestor. During this period, the "book of law", believed to be Deuteronomy was 'found' in the Temple during renovations. Scholars say that the D source (the Deuteronomist) lived during the reign of Josiah and not only completed much of the Pentateuch, but also wrote the Deuteronomic history - the Books of Joshua, Judges, 1 and 2 Samuel and 1 and 2 Kings. These reforms, the largely successful permanent reintroduction of monotheism and the substantial completion of major works of the Bible, could be regarded as the origin of Judaism - late in the seventh century BCE.
We can identify changes to the theology of the Bible, starting during the Babylonian exile of the sixth century BCE. If we regard the new ideas absorbed during this period as essential to the definition of Judaism, then the start date of Judaism could be 500 BCE or later.
Christianity
It is universally accepted that Christianity began in the middle of the first century CE.
Judaism is older
1) If we accept the traditional date for the life of Abraham, Judaism is about 2000 years older than Christianity.
2) If we accept the traditional date for the life of Moses, Judaism is about 1400 years older than Christianity.
3) If we accept the scholarly research and the findings of archaeologists, Judaism is about 500 - 600 years older than Christianity.
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