Freedom is not free. There is a price to pay.
When God spared the Jews from the death of the firstborn, the price was and still is that all firstborns, bechorot, belong to God. “On the day that I struck down every firstborn in the land of Egypt, I sanctified every firstborn in Israel for Myself, from man to beast they shall be Mine.”
There are three categories of bechorot.
The first is the human bechor. Every son, firstborn to his mother, belongs to and has to be redeemed by God’s representative, the kohen, through the pidyon haben ceremony in which the father, or the bechor himself, purchases the firstborn back from the kohen for 5 sela’im coins. The determining factor as to whether a person is a bechor requiring pidyon haben is if he is the firstborn of his mother, not his father.
Accordingly, a son first born to a woman who married a man who already had a son from a previous marriage is a bechor for the purpose of requiring pidyon haben even though he is not a bechor for the purposes of receiving a double portion in his father’s estate for which privilege, the determining factor is being the firstborn of the father.
The second type of bechor is the firstborn offspring of certain kosher animals. The male cow, sheep or goat firstborn to its mother belongs to the kohen and is known as bechor behemah tehorah. It acquires the status of bechor at birth and, unlike the human bechor who is redeemed, the bechor behemah tehorah is actually offered on the outer altar as a korban. Contrary to other animals that have to be declared holy before they are offered as a sacrifice to God, the bechor behemah tehorah automatically assumes the status of an offering, a korban, without having to be declared holy, even though there is a mitzvah for its owner to do so.
The korban bechor behemah tehorah belongs to the less holy category of kodashim kalim, and is therefore free from the many restrictions that apply to the holier forms of sacrifice known as kodshei kodashim, holy of holies. Accordingly, the korban bechor may be slaughtered anywhere within the Temple courtyard and not exclusively in the northern part. The blood of the korban bechor was applied by sprinkling it once on the altar. Except for those parts, (the fat, the kidneys and the liver known as the eimurim) which were burned on the altar, the korban bechor was eaten entirely and exclusively by the kohanim.
Unlike kodhsei kodashim that could only be eaten within the confines of the Temple Courtyard, the korban bechor could be eaten anywhere in Jerusalem. Unlike kodshei kodashim, which could only be eaten during the day of its slaughter until midnight, the korban bechor could be eaten during the day of its slaughter, throughout the following night and through the next day until nightfall.
As soon as it is born, the bechor behemah tehorah, even today, is automatically considered kodesh, sanctified to the kohen. As such, it may not be slaughtered for food, shorn for its wool or worked in any way. It can only be used as a korban. This presents a problem. Today, in the absence of the Temple, the animal cannot be sacrificed. It would have to be left out to pasture. But the risk of somebody erring and using this sanctified animal for secular use is high. Accordingly, the rabbis have devised a way whereby the bechor behemah tehorah today, will not be born in its sanctified state.
This is achieved by selling a part ownership in the mother, before the birth of the firstborn, to a non-Jew. Since it is only the firstborn of an animal owned solely by a Jew that renders the firstborn sanctified, it follows that if the mother is partially owned by a non-Jew, the firstborn is not considered holy at birth and is free for secular use. In view of the fact that there is a difference of opinion among the Rishonim as to how a non-Jew acquires from a Jew property in an animal, (according to Rashi, the payment of money alone is sufficient and according to Rabbeinu Tam the animal must be pulled along) the Rema requires that the non-Jew both pay money and pull the animal along.