What is the greatest common divisor (GCD)?

The greatest common divisor (GCD) is defined as the largest integer that can divide two or more numbers without leaving a remainder. This concept is essential in number theory and is used in various applications, including simplifying fractions and finding common denominators.

When determining the GCD, the focus is on identifying the largest number that each of the numbers in the set can be divided by evenly. For instance, if you consider the numbers 12 and 15, the GCD is 3 because 3 is the largest integer that divides both numbers completely (12 ÷ 3 = 4 and 15 ÷ 3 = 5).

In contrast, the other options do not accurately reflect the definition of the GCD. The smallest positive integer refers to the number 1, which is not necessarily the greatest common divisor. The highest number in a set does not take into account the divisibility aspect and simply refers to the maximum value present. Lastly, the average of two numbers is a calculated mean and has no relation to divisibility or the commonality between numbers. Thus, the choice that accurately describes the GCD is indeed the largest integer that divides two numbers.