What is "prime factor"

Prime \Prime\, a. [F., fr. L. primus first, a superl. corresponding to the compar. prior former. See Prior, a., Foremost, Former, and cf. Prim, a., Primary, Prince.]

1. First in order of time; original; primeval; primitive; primary. ``Prime forests.''
   --Tennyson.

   She was not the prime cause, but I myself.
   --Milton.

   Note: In this sense the word is nearly superseded by primitive, except in the phrase prime cost.

2. First in rank, degree, dignity, authority, or importance; as, prime minister. ``Prime virtues.''
   --Dryden.

3. First in excellence; of highest quality; as, prime wheat; a prime quality of cloth.

4. Early; blooming; being in the first stage. [Poetic]

   His starry helm, unbuckled, showed him prime In manhood where youth ended.
   --Milton.

5. Lecherous; lustful; lewd. [Obs.]
   --Shak.

6. Marked or distinguished by a mark (') called a prime mark.

7. (Math.)

   1. Divisible by no number except itself or unity; as, 7 is a prime number.

   2. Having no common factor; -- used with to; as, 12 is prime to 25. Prime and ultimate ratio. (Math.). See Ultimate. Prime conductor. (Elec.) See under Conductor. Prime factor (Arith.), a factor which is a prime number. Prime figure (Geom.), a figure which can not be divided into any other figure more simple than itself, as a triangle, a pyramid, etc. Prime meridian (Astron.), the meridian from which longitude is reckoned, as the meridian of Greenwich or Washington. Prime minister, the responsible head of a ministry or executive government; applied particularly to that of England. Prime mover. (Mech.)

      1. A natural agency applied by man to the production of power. Especially: Muscular force; the weight and motion of fluids, as water and air; heat obtained by chemical combination, and applied to produce changes in the volume and pressure of steam, air, or other fluids; and electricity, obtained by chemical action, and applied to produce alternation of magnetic force.

      2. An engine, or machine, the object of which is to receive and modify force and motion as supplied by some natural source, and apply them to drive other machines; as a water wheel, a water-pressure engine, a steam engine, a hot-air engine, etc.

   3. Fig.: The original or the most effective force in any undertaking or work; as, Clarkson was the prime mover in English antislavery agitation.

      Prime number (Arith.), a number which is exactly divisible by no number except itself or unity, as 5, 7, 11.

      Prime vertical (Astron.), the vertical circle which passes through the east and west points of the horizon.

      Prime-vertical dial, a dial in which the shadow is projected on the plane of the prime vertical.

      Prime-vertical transit instrument, a transit instrument the telescope of which revolves in the plane of the prime vertical, -- used for observing the transit of stars over this circle.

n. A factor of a given integer which is also a prime number. vb. (context transitive English) to reduce an integer to its set of prime factors

n. the prime factors of a quantity are all of the prime quantities that will exactly divide the given quantity

In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicities; the process of determining these factors is called integer factorization. The fundamental theorem of arithmetic says that every positive integer has a single unique prime factorization.

To shorten prime factorizations, factors are often expressed in powers (multiplicities). For example,

360 = 2 × 2 × 2 × 3 × 3 × 5 = 2 × 3 × 5, 
in which the factors 2, 3 and 5 have multiplicities of 3, 2 and 1, respectively.

For a prime factor p of n, the multiplicity of p is the largest exponent a for which p divides n exactly.

For a positive integer n, the number of prime factors of n and the sum of the prime factors of n (not counting multiplicity) are examples of arithmetic functions of n that are additive but not completely additive.