Definition

Noun: The largest positive integer that is a divisor of all numbers in a given set of integers. It is the greatest number that can exactly divide each of the numbers in the set, leaving no remainder.

Usage

The term "highest common factor" is used primarily in mathematics, specifically in number theory and arithmetic. It is applied when finding the greatest shared divisor of two or more integers.

Examples

• The highest common factor of 12 and 18 is 6.
• To simplify the fraction 24/36, you first find the highest common factor of 24 and 36, which is 12.
• The numbers 7 and 15 have a highest common factor of 1, meaning they are co-prime.

Advanced Usage

• The concept of the highest common factor is fundamental to the Euclidean algorithm, an efficient method for computing it.
• In algebra, finding the highest common factor of the terms in a polynomial expression is a key step in factoring.

Variants and Related Words

• Greatest Common Divisor (GCD): This is the most common synonym and is used interchangeably with "highest common factor."
• Greatest Common Factor (GCF): Another exact synonym, frequently used in educational contexts.

Synonyms

• Greatest Common Divisor (GCD)
• Greatest Common Factor (GCF)

Related Phrases and Concepts

• Lowest Common Multiple (LCM): The smallest positive integer that is a multiple of all numbers in a given set. The HCF/GCD and LCM of two numbers are related: HCF(a, b) × LCM(a, b) = a × b.
• Relatively Prime / Co-prime: A term describing two or more numbers whose highest common factor is 1.

Noun

1. the largest integer that divides without remainder into a set of integers