What is the definition of HCF?

HCF: Highest Common Factor

The highest common factor (HCF) of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder.

For example, the HCF of 12 and 18 is 6, since 6 is the largest positive integer that divides both 12 and 18 without leaving a remainder.

The HCF can be found using a variety of methods, including the Euclidean algorithm and the prime factorization method.

Euclidean algorithm

The Euclidean algorithm is a method for finding the HCF of two numbers by repeatedly dividing the larger number by the smaller number and taking the remainder. The HCF is the last non-zero remainder.

For example, to find the HCF of 12 and 18, we can use the Euclidean algorithm as follows:

1. Divide 18 by 12: 18 = 12 * 1 + 6

2. Divide 12 by 6: 12 = 6 * 2 + 0

The last non-zero remainder is 6, so the HCF of 12 and 18 is 6.

Prime factorization method

The prime factorization method involves writing each number as a product of its prime factors. The HCF is then the product of the common prime factors, raised to the lowest power they appear in either number.

For example, to find the HCF of 12 and 18, we can write them as follows:

12 = 2 * 2 * 3

18 = 2 * 3 * 3

The common prime factors are 2 and 3, so the HCF of 12 and 18 is 2 * 3 = 6.

The HCF of two numbers can be used to find the least common multiple (LCM) of those numbers. The LCM is the smallest positive integer that is divisible by both numbers.

The LCM of two numbers can be found by multiplying the HCF of those numbers by the product of the two numbers.

For example, to find the LCM of 12 and 18, we can use the HCF and the product of the two numbers as follows:

HCF of 12 and 18 = 6

Product of 12 and 18 = 12 * 18 = 216

LCM of 12 and 18 = 6 * 216 = 1296