Factor Pairs
Factor pairs are pairs of whole numbers that, when multiplied together, yield a specific product. Each product has a unique set of factor pairs, and because multiplication is a commutative operation, the order of the factors does not affect the result. For example, the numbers 3 and 8 form a factor pair for the product 24, as both 3 × 8 and 8 × 3 equal 24. This process of identifying factor pairs is known as factorization and serves as a foundational concept in mathematics, paving the way for more advanced topics such as determining greatest common factors and prime factorization.
Understanding factor pairs is essential for simplifying fractions and performing various arithmetic operations. For instance, recognizing that 3 is a factor of 24 can make it easier to simplify fractions involving the number 24. Additionally, all whole numbers greater than 1 can be categorized as either prime numbers, which have only the trivial factor pair of 1 and themselves, or composite numbers, which possess nontrivial factor pairs. Familiarity with factor pairs is beneficial for anyone looking to enhance their mathematical skills and tackle more complex concepts in mathematics.
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Subject Terms
Factor Pairs
The two numbers that operate on each other through the operation of multiplication are called factors, and the resulting solution of the multiplication operation is called the product of these two factors. Two factors that multiply together to yield a given whole-number product are called a factor pair of that product.
Because multiplication is a commutative operation, the order of factors can be changed without obtaining a different result. For this reason, a given factor pair uniquely defines a given product. Also, for a given product, identifying one factor uniquely determines the second factor within the factor pair.
For any whole number, the ability to identify factor pairs of that number can prove useful. This process of identifying factor pairs of a whole number is the most basic process known as factorization, and it is the foundation for increasingly complex arithmetic work involving multiplication (and including determining greatest common factors, prime factorization, and reducing fractions). Familiarity with this basic concept is implicit when moving on to more advanced concepts such as factoring and simplifying algebraic expressions, and to more specialized fields of mathematics, such as number theory (and including the fundamental theorem of arithmetic, first proposed by the Greek mathematician Euclid).
Overview
Whole number factors within a multiplication expression operate upon each other to uniquely determine a whole number product. Multiplication, like addition but in contrast to multiplication and division, is a commutative operation, which means that changing the sequence of numbers within the operation does not alter the final solution.
The process of factorization begins by being able to take a given product and identify its factor pairs. Given the number 24, knowledge of 3 × 8 = 24 = 8 × 3 indicates that 3 and 8 are a factor pair of 24.
Identifying factor pairs of whole numbers is a ubiquitous skill within mathematics. When one is simplifying a rational number in fraction form, a common approach is to look at the whole numbers in the numerator and denominator and determine if one of them is a factor of the other, so that this factor can be eliminated.
If one is aware that 3 and 8 are a factor pair of 24, this solution is virtually immediate. Indeed, in this case, it is not necessary to exhaustively consider all possible factor pairs, because the presence of the 3 in the numerator indicates which factor pair of 24 is needed, as opposed to the factor pair 4 and 6 or 12 and 2. If a common factor is not immediately visible, then one must factor both the numerator and the denominator to locate a common factor.
Due to the identity property of multiplication, it is the case that 1 × m = m for any value of m. Therefore, any number m will have the trivial factor pair of 1 and m. Whole numbers greater than 1 with no factor pairs other than this are classified as prime numbers. Whole numbers greater than 1 that do have nontrivial factor pairs are called composite numbers.
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