Hello and thank you for your question,
During the routine gynecological exam you get a routine PAP smear, however, if you request to be checked for STDs, it can be easily done at the same time. Gonorrhea and Chlamydia can be checked with a culture during the exam and other STDs, such as HIV, Herpes, Syphilis, can be tested with a blood test. Unless you ask to be tested for STDs, it will not be done.
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Yes, that is exactly how it works. They look for antibodies on the blood test and during a swab (for gonorrhea/chlamydia), they look for the actual bacteria.
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