HCF (GCF)

The Highest Common Factor (HCF), also known as the Greatest Common Factor (GCF), is used in the field of mathematics to simplify fractions and understand ratios. By dividing the given numbers by their HCF, we can provide an equivalent fraction in its simplest form. This HCF can be evaluated for two or more numbers.
For example, the highest common factor of 36 and 48 is 12; it is the largest number that can divide both 36 and 48 exactly.
 

This mathematical concept can be traced back to ancient Babylonian mathematics (circa 1800 BCE). The Babylonians used HCF algorithms to solve problems involving numbers with fractions and divisors in their calculations. The use of HCF was then formalized by the Greek mathematician Euclid in his book, Elements. 
 

The Highest Common Factor helps us find the largest number that divides evenly into both. Knowing its properties makes it easier to solve the problems. Let’s discuss some of the properties of HCF.


 

• Divisibility property: The HCF of two or more numbers always divides each of the given numbers exactly.
• Relation with Numbers: The HCF of two numbers is always less than or equal to the smallest number.
• HCF of prime numbers: If we have two different prime numbers, their HCF is always 1.
• If one number divides the other: The HCF will be a smaller number, if one number divides the other number completely.
• HCF of equal numbers: The HCF is the number itself if the two numbers are equal.
• HCF of consecutive numbers: The HCF of any two consecutive natural numbers is always 1.
• HCF with zero:
  HCF (a, 0) = a (where a ≠ 0)
  HCF (0, 0) is undefined.
• Multiplicative relationship with LCM: For any two positive integers a and b:    HCF (a, b) × LCM (a, b) = a × b
• HCF of consecutive even numbers: The HCF of two consecutive even numbers is always 2.
• Product of HCF and Co-prime numbers: If two numbers are co-prime (HCF = 1), then their product is equal to their LCM. 

The HCF (Highest Common Factor) can be found using methods like Prime Factorization, which identifies and multiplies common prime factors, Division, which repeatedly divides numbers until the remainder is zero, and Listing Factors, which compares all factors to find the greatest one. Here we have given three different methods to find the HCF of numbers:
 

Prime factorization is the process of expressing a number as the product of its prime factors, which are the smallest prime numbers that multiply together to equal the original number. To find the HCF of the given numbers, we need to find the common prime factors. For example, to find the HCF of 200 and 300, prime factorize them both:


Prime Factorization of 200 =  2³ × 5²
Prime Factorization of 300 = 2² × 3 × 5²

The common factors among both are: 2, 2, 5, 5
The HCF of 200 and 300 = 2 × 2 × 5 × 5 = 100 
Thus, the HCF is 100. 

The division method for finding the HCF involves dividing the larger number by the smaller number, then using the remainder as the new divisor, and repeating the process until the remainder becomes zero. The last non-zero divisor is the HCF. For example, to find the HCF of 200 and 300, divide the larger number by the smaller number:


That is, 300 ÷ 200 = 1 (quotient) and the remainder as 100.
Next, divide replacing the larger number (300) with the smaller number (200), and the smaller number (200) with the remainder (100), repeating until the remainder is 0, where the last non-zero number is the HCF.


That is, divide 200 by 100:

200 ÷ 100 = 2 (quotient) and the remainder 0.
Since we got 0 as the remainder, 100 is the HCF.
Thus, the HCF of 200 and 300 is 100.

In this method, the factors of both the numbers are listed, thereby manually checking the common factors among them and then finding the highest common factor from the lot. For example, let’s take 200 and 300:


Factors of 200 = 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200.

Factors of 300 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, and 300.

Common Factors of 200 and 300 = 1, 2, 4, 5, 10, 20, 25, 50, 100

The highest common factor from the common factors = 100

Hence, the HCF of 200 and 300 is 100.
 

While finding the HCF of numbers, using these tips and tricks will help you reach the answers faster. Practice identifying these patterns and shortcuts below.

 

• If there are more than two numbers to find the HCF, first find the HCF between two numbers, and then find the HCF of the third number using the result of the first two numbers.
• The HCF of co-prime numbers is always 1. Co-prime numbers are numbers that have no common factors other than 1. Finding HCF among prime numbers is far easier and faster.
• If one number is a factor of another number, then the smaller number is the HCF of both numbers.