Definition

1. Noun:
   • A prime number that divides a given integer exactly: A "prime factor" is a prime number that is a factor of a given number, meaning it divides that number without leaving a remainder. Every integer greater than 1 is either a prime number itself or can be expressed uniquely as a product of prime factors.

Usage

• The term is used in mathematics, specifically in number theory and arithmetic, to describe the fundamental building blocks of integers.
• It is typically used in the plural form "prime factors" when discussing the complete set for a given number.
• Common structures: "The prime factors of X are...", "to find the prime factors", "to decompose into prime factors".

Examples

• Noun:
  • The prime factors of 12 are 2 and 3, because 12 = 2 × 2 × 3.
  • Finding the prime factors of a large number can be computationally difficult.
  • In the equation, the number was broken down into its constituent prime factors.

Advanced Usage

• "Prime factorization": The process of decomposing a composite number into a product of its prime factors.
  • The prime factorization of 30 is 2 × 3 × 5.
• "Distinct prime factors": Refers to the unique prime numbers in the factorization, ignoring their multiplicity (how many times they appear).
  • The number 12 has two distinct prime factors: 2 and 3.

Variants and Related Words

• Factor (noun): A number or quantity that divides another number exactly.
  • 3 is a factor of 15.
• Prime (noun/adjective): A whole number greater than 1 that has no positive divisors other than 1 and itself.
  • 7 is a prime number.
• Factorization (noun): The process of writing a number or expression as a product of factors.
  • We studied the factorization of polynomials.

Synonyms

• Prime divisor: A prime number that divides another number exactly. (Very close in meaning, often used interchangeably in context.)

Related Concepts (Not Phrasal Verbs or Idioms)

• Fundamental Theorem of Arithmetic: The theorem that states every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This theorem underpins the concept of prime factors.
• Greatest Common Divisor (GCD): The largest number that divides two or more numbers. Found using their prime factorizations.
• Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers. Found using their prime factorizations.

Noun

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