The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of a set of numbers is the largest number that divides each of them without leaving a remainder.

To find the HCF of 12, 16, and 28, we will use the Prime Factorization Method.

Step 1: Find the Prime Factors of Each Number

Break down each number into its prime factors (prime numbers that multiply together to make the original number).

$12 = 2 \times 2 \times 3 = 2^{2} \times 3$
$16 = 2 \times 2 \times 2 \times 2 = 2^{4}$
$28 = 2 \times 2 \times 7 = 2^{2} \times 7$

Step 2: Identify the Common Prime Factors

Look at the prime factors of all three numbers and find the factors that are common to all of them.

The only common prime factor is 2.

Step 3: Choose the Lowest Power of the Common Factors

For the common prime factor (2), we look at the exponents (powers) from each factorization: 2² (from 12 and 28) and 2⁴ (from 16). The lowest power is 2².

Step 4: Multiply the Common Factors with the Lowest Powers

Multiply these selected factors together to get the HCF.

$HCF = 2^{2} = 2 \times 2 = 4$

Therefore, the HCF of 12, 16, and 28 is 4.

Related Topics & Formulae

Related Topic: Least Common Multiple (LCM)
The LCM of numbers is the smallest number that is a multiple of each of the numbers. While HCF deals with common divisors, LCM deals with common multiples.

Formula Connection:
For any two numbers, a relationship exists between their HCF and LCM:
$HCF (a, b) \times LCM (a, b) = a \times b$
This formula is a quick way to find one if you know the other.