Factor

A factor is a portion of a quantity, usually an integer or polynomial that, when multiplied by other factors, gives the entire quantity. The determination of factors is called factorization (or sometimes "factoring").

In number theoretic usage, a factor of a number  is equivalent to a divisor of  (Ore 1988, p. 29; Burton 1989, p. 26). The divisors of a number  are given in the Wolfram Language by the command Divisors[n]. In elementary education, the term "factor" is sometimes used to mean proper divisor, i.e., a factor of  other than the number  itself. However, as a result of the confusion this practice creates and its inconsistency with the mathematical literature, it should be discouraged.

It is usually desired to break factors down into the smallest possible pieces so that no factor is itself factorable. For integers, the determination of such prime factors is called prime factorization. For large integers, the determination of all factors is usually very difficult except in exceptional circumstances.

The term "factor" is occasionally misused, including by no less "authority" than The New York Times, where Fox (2006) wrote, "He was 88, which can be factored as 1, 2, 4, 8, 11, 22, 44, and 88." This usage is incorrect since the given numbers are indeed factors, but the collection of factors does not comprise a factorization.

REFERENCES:

Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989.

Fox, M. "George Lencher, 88, Dies After Life by the Numbers." The New York Times. Obituaries. May 14, 2006.

Ore, Ø. Number Theory and Its History. New York: Dover, 1988.