Etymology
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factor (v.)
1610s, "act as an agent, manage," from factor (n.). The use in mathematics is attested from 1837. Related: Factored; factoring.
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prime (n.)

"earliest canonical hour of the day" (6 a.m.), from Old English prim and Old French prime and directly from Medieval Latin prima "the first service," from Latin prima hora "the first hour" (of the Roman day), from Latin primus "first, the first, first part" (see prime (adj.)).  (In classical Latin, the noun uses of the adjective meant "first part, beginning; leading place.")

By extension, "the first division of the day" 6 a.m. to 9 a.m. (early 13c.). The sense of "beginning of a period or course of events" is from late 14c. From the notion of "the period or condition of greatest vigor in life" (by 1530s) comes the specific sense "springtime of human life" (often meaning ages roughly 21 to 28) is from 1590s. Also from 1590s as "that which is best in quality, highest or most perfect state of anything." As "a prime number," by 1530s.

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factor (n.)

early 15c., "commercial agent, deputy, one who buys or sells for another," from French facteur "agent, representative" (Old French factor, faitor "doer, author, creator"), from Latin factor "doer, maker, performer," in Medieval Latin, "agent," agent noun from past participle stem of facere "to do" (from PIE root *dhe- "to set, put"). In commerce, especially "a commission merchant." Mathematical sense ("The Quantities given to be multiplied one by the other are called Factors") is from 1670s. Sense of "circumstance producing a result" is attested by 1816, from the mathematical sense.

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prime (v.)

"to fill, charge, load" (a weapon, before firing), 1510s, probably from prime (adj.). General sense of "perform the first operation on, prepare (something, especially wood, etc., for painting)" is from c. 1600. To prime a pump (1769) meant to pour water down the tube, which saturated the sucking mechanism and made it draw up water more readily. Related: Primed; priming.

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prime (adj.)

late 14c., "first, original, first in order of time," from Old French prime and directly from Latin primus "first, the first, first part," figuratively "chief, principal; excellent, distinguished, noble" (source also of Italian and Spanish primo), from Proto-Italic *prismos, superlative of PIE *preis- "before," from root *per- (1) "forward," hence "in front of, before, first, chief."

The meaning "of fine quality, of the first excellence" is from c. 1400. The meaning "first in rank, degree, or importance" is from 1610s in English. Arithmetical sense (as in prime number, one indivisible without a remainder except by 1) is from 1560s; prime meridian "the meridian of the earth from which longitude is measured, that of Greenwich, England," is from 1878. Prime time originally (c. 1500) meant "spring time;" the broadcasting sense of "peak tuning-in period" is attested by 1961.

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prime minister 

"leading minister of a government, the chief of the cabinet or ministry," 1640s, see prime (adj.) and minister (n.). Applied to the First Minister of State of Great Britain since 1694.

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Rh factor 
1942, from the first letters of rhesus; so called because the blood group, and its effects, were discovered in the blood of rhesus monkeys (1941).
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factorial (n.)
1816, in mathematics, from factor + -al (2). As an adjective from 1837 in mathematics; from 1881 as "pertaining to a factor."
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subprime (adj.)
also sub-prime, of loans, etc., by 1978, in frequent use from 1996, from sub- + prime (adj.).
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multifactorial (adj.)

also multi-factorial, "involving or dependent on a number of factors," 1920, from multi- "many" + factor (n.) + -ial.

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