The greatest common divisor (gcd) of two integers is the largest positive integer that divides both of them without leaving any remainder. Let's consider two consecutive integers, n and n + 1, where n is any integer.

Now, let's find the gcd of n and n + 1:

Step 1: Assume d is the gcd of n and n + 1.

Step 2: Since d divides n, there exists an integer k such that n = d * k.

Step 3: Now, let's examine n + 1: n + 1 = d * k + 1

Step 4: Since d divides both n and n + 1, it must also divide their difference: (n + 1) - n = (d * k + 1) - (d * k) = 1

Step 5: Since d divides 1 without leaving any remainder (d * 1 = 1), the gcd of n and n + 1 (d) must be 1.

Hence, gcd(n, n + 1) = 1 for any consecutive integers n and n + 1. This property holds true for all integer values of n, whether positive, negative, or zero.