Let $m$ be a positive integer, find the greatest common divisor of $m$ and $m+2$ .

Step 1: Identify the factors of each number Let $m$ be a positive integer. To find the factors of $m$ , we will assume that it ranges from 1 to $m$ . Similarly, for $m+2$ , the factors will range from 1 to $m+2$ . Note that the prime factors of $m+2$ can only be greater than 2 (otherwise $(m+2)$ would be an even number, and our given $m$ is odd). Step 2: Compare the factors between the two numbers We want to find the greatest common divisor (GCD) of the two numbers, which requires comparing the factors we found in Step 1. If both numbers are even or both numbers are odd, they share at least one common factor of 1. For even numbers, they both have a factor of 2. Step 3: Find the greatest common divisor In the case of odd numbers, the only possible common factor between $m$ and $m+2$ is 1. This is because they are consecutive odd numbers and are not divisible by any other common even factor. Thus, the GCD is 1 for any pair of consecutive odd numbers, i.e., $m$ is odd. In the case of even numbers, since both $m$ and $m+2$ are divisible by 2, they share a common factor of 2. No other even factors are shared by both numbers as they are consecutive even numbers (similar to the consecutive odd numbers). Thus, the GCD is 2 for any pair of consecutive even numbers, i.e., $m$ is even. Therefore, the greatest common divisor of $m$ and $m+2$ is 1 if $m$ is odd, and 2 if $m$ is even.