HCF

HCF stands for highest common factor, the greatest positive integer that divides two or more integers without leaving a remainder. It is the same concept as the greatest common divisor (GCD).

HCF can be found by prime factorization or by the Euclidean algorithm. In prime factorization, write each number as a product of primes and take the product of the primes that appear in both factorizations with the smallest exponents. In the Euclidean method, repeatedly apply the division algorithm: gcd(a,b) = gcd(b, a mod b), until the remainder is zero.

Examples illustrate the idea. For 48 and 18, gcd(48, 18) = 6, since 48 = 2^4 × 3 and 18 = 2 × 3^2, with the common factors 2 and 3 giving 2 × 3 = 6. Another example: gcd(27, 14) = 1, meaning the numbers are coprime.

Properties include that the HCF is nonnegative and gcd(a,b) = gcd(b,a). Also gcd(a,0) = |a|, and gcd(0,0) is undefined in some contexts. The gcd of multiple numbers can be extended as gcd(a,b,c) = gcd(gcd(a,b), c).

Relation to other concepts: gcd(a,b) × lcm(a,b) = |a × b| for nonzero a and b, linking the greatest common factor with the least common multiple. The HCF is used to simplify fractions, solve Diophantine equations, and determine common divisors. The concept extends to polynomials, where the greatest common factor is the highest-degree polynomial that divides each given polynomial.