"the primary purposes" is a correct and usable phrase in written English.
It can be used when discussing the main or most important reasons or goals for something. Example: The primary purposes of this project are to increase efficiency and reduce costs for our company.
The primary purposes of this flight are to rotate the crew, deliver supplies and take away trash and discarded equipment.
Maps are therefore often classified by the primary purposes they serve.
The primary purposes of the movements of the small intestine are to provide mixing and transport of intraluminal contents.
Isn't financial security in the face of catastrophic medical expenses one of the primary purposes of health insurance?
The primary purposes of this flight are to rotate the crew, fill the larder, change the linens and take out the garbage.
But finding and nurturing the talented people who can advance our knowledge must surely be one of the primary purposes of university.
Finally, unlike civilian investments, defense-related investments fulfill one of the primary purposes of the federal government: to keep our nation secure.
Spring Lake and Ocean Grove have tours of their fancy inns, while Keyport and Freehold do mainly houses, one of the primary purposes of which is to bring people to shop in downtown districts.
Petitioners propose several ways in which the narcotics-detection purpose of the instant checkpoint program may instead resemble the primary purposes of the checkpoints in [Michigan v.] Sitz (1990) and [United States v.] Martinez-Fuerte (1976).
One of the primary purposes of the tribunal and order is to attempt/contribute to seize political and military control of Western European multiculturalist regimes and to try, judge and punish Western European cultural Marxist/multiculturalist perpetrators (category A, B and C traitors) for crimes committed against the indigenous peoples of Europe from 1955 until this day".
And in New York and especially during Fashion Week, Ms. Coles, expanding on the subject, was saying, you can easily form an impression that the majority of people have altogether forgotten that one of the primary purposes of looking good in clothes is to attract the amatory interest of someone who might wish to help you remove them.
