A geographer is analyzing data on coastal erosion and finds that the rate of erosion (in meters per year) for different coastal regions can be represented by a set of integers. If the greatest common divisor (GCD) of these rates for three regions is 6 and the total erosion rate for these regions over one year is 84 meters, what is the maximum possible erosion rate for any one region?

["Let the erosion rates for the three coastal regions be $ a $, $ b $, and $ c $, all positive integers. We are told:", "- $ \gcd(a, b, c) = 6 $\n- $ a + b + c = 84 $\n- We are to find the maximum possible value of any one of $ a, b, c $, given these constraints.", "Since the greatest common divisor is 6, we can write each rate as a multiple of 6:", "$$\na = 6m, \quad b = 6n, \quad c = 6p\n$$", "where $ m, n, p $ are positive integers and $ \gcd(m, n, p) = 1 $ (since we've factored out the GCD).", "Substituting into the sum:", "$$\n6m + 6n + 6p = 84 \Rightarrow 6(m + n + p) = 84 \Rightarrow m + n + p = 14\n$$", "We now seek to maximize one of $ a, b, c $, which corresponds to maximizing the largest among $ 6m, 6n, 6p $. This occurs when two of $ m, n, p $ are as small as possible (to allow the third to be as large as possible), while still satisfying $ \gcd(m, n, p) = 1 $.", "The smallest positive integers for two of the variables are 1 and 1 (since $ m, n, p $ are positive integers). Try $ m = 1 $, $ n = 1 $, then:", "$$\nm + n + p = 1 + 1 + p = 14 \Rightarrow p = 12\n$$", "Now check $ \gcd(m, n, p) = \gcd(1, 1, 12) = 1 $, which satisfies the condition.", "Then the corresponding erosion rates are:", "- $ a = 6 \cdot 1 = 6 $\n- $ b = 6 \cdot 1 = 6 $\n- $ c = 6 \cdot 12 = 72 $", "Sum: $ 6 + 6 + 72 = 84 $, correct.", "GCD of 6, 6, 72 is 6 — valid.", "Can we get a larger value? Try $ m = 1 $, $ n = 2 $ → $ p = 11 $, then max is $ 6 \cdot 11 = 66 < 72 $. Any try with $ p > 12 $ would require $ m + n < 2 $, but $ m, n \geq 1 $ implies $ m + n \geq 2 $, so $ p \leq 12 $. Thus 72 is the maximum possible.", "Therefore, the maximum possible erosion rate for any one region is $ \boxed{72} $ meters."]