Given the equations
\[ N = 5a + 3b + 5c \] and \[ N = 4a + 5b + 4c, \]
we can set them equal to each other as follows:
\[ 5a + 3b + 5c = 4a + 5b + 4c. \]
Rearranging this equation, we get:
\[ 5a - 4a + 3b - 5b + 5c - 4c = 0, \] which simplifies to:
\[ a - 2b + c = 0. \]
From this, we can express \( c \) in terms of \( a \) and \( b \):
\[ c = 2b - a. \]
Substituting this back into \( N \), we find:
\[ N = 5a + 3b + 5(2b - a) = 5a + 3b + 10b - 5a = 13b. \]
Thus we have:
\[ N = 13b. \]
Given the condition \( 131 < N < 150 \), substituting for \( N \) yields:
\[ 131 < 13b < 150. \]
Dividing through by 13, we find:
\[ \frac{131}{13} < b < \frac{150}{13}, \]
which simplifies to:
\[ 10.0769 < b < 11.5385. \]
Since \( b \) must be a distinct positive integer, the only possible value for \( b \) is \( 11 \).
Now substituting \( b = 11 \) back into the expression for \( N \):
\[ N = 13 \times 11 = 143. \]
Now substituting \( b = 11 \) into the equation \( c = 2b - a \):
\[ c = 2 \times 11 - a = 22 - a. \]
Next, we need to ensure that \( a \), \( b \), and \( c \) are distinct positive integers. We have \( b = 11 \) and \( c = 22 - a \).
To keep \( c \) positive, we require:
\[ 22 - a > 0 \implies a < 22. \]
To maintain distinctness among \( a \), \( b \), and \( c \), we require:
- \( a \neq 11 \) (since \( b = 11 \)),
- \( a \neq 22 - a \) (i.e., \( 2a \neq 22 \implies a \neq 11 \)).
Thus, we can choose \( a \) as any integer from 1 to 21, excluding 11.
Let's evaluate \( a + 2b + 3c \):
\[ a + 2b + 3c = a + 2(11) + 3(22 - a) = a + 22 + 66 - 3a = -2a + 88. \]
To maximize \( a + 2b + 3c \), we should minimize \( a \). The smallest value \( a \) can take, that is a distinct positive integer, is 1.
Substituting \( a = 1 \):
\[ a + 2b + 3c = -2(1) + 88 = 86. \]
We can verify with larger values of \( a \):
- If \( a = 2 \): \( -2(2) + 88 = 84 \)
- If \( a = 3 \): \( -2(3) + 88 = 82 \)
- Continuing this way increases \( -2a \), thus lowering the total.
The maximum achievable is when \( a = 1\):
Therefore, the maximum possible value of \( a + 2b + 3c \) is
\[ \boxed{86}. \]
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