Greatest common divisor

The greatest common divisor (often called GCD or HCF for highest common factor) is a number that divides two or more whole numbers evenly. It is called “greatest” because it is the largest number that can divide all the given numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, since 6 is the biggest number that divides both 12 and 18 evenly.

To find the GCD, we look for all the factors of each number and then find the largest one they share. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. Both have 1, 2, 3, and 6 as common factors, but the greatest of these is 6 — the GCD.

There are several ways to find the GCD. One way is by listing factors, as shown above. Another is by prime factorization. This means breaking each number into its prime factors and multiplying the ones they share. For example, 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Both share 2 and 3, and multiplying them gives 6, which is the GCD. A third method is using the Euclidean algorithm, a simple process that uses division and remainders to find the GCD quickly.

The GCD is useful in many parts of mathematics. It helps simplify fractions by dividing both the numerator and denominator by their greatest common divisor. For example, to simplify 18/24, we divide both numbers by 6, giving 3/4. The GCD is also used in solving number problems, ratios, and equations where relationships between whole numbers are important.

In real life, the GCD can help with tasks like dividing objects into equal groups or comparing quantities. For example, if 12 red balloons and 18 blue balloons need to be divided equally among children, the GCD of 12 and 18 (which is 6) shows that 6 children can each get the same number of balloons.

The idea of the greatest common divisor shows how numbers can be compared based on what they share. It is one of the building blocks of arithmetic and number theory, helping us understand how numbers connect and how they can be simplified.