To solve the problem, we need to find the coefficients \( a, b, c, d \) of the plane equation \( ax + by + cz = d \) that passes through the point \( (2, 3, -5) \) and is perpendicular to the given planes \( 2x + y - 5z = 10 \) and \( 3x + 5y - 7z = 12 \).
### Step 1: Find the normal vectors of the given planes
The normal vector of the first plane \( 2x + y - 5z = 10 \) is \( \vec{n_1} = (2, 1, -5) \).
The normal vector of the second plane \( 3x + 5y - 7z = 12 \) is \( \vec{n_2} = (3, 5, -7) \).
### Step 2: Find the direction ratios of the required plane
Since the required plane is perpendicular to both given planes, its normal vector \( \vec{n} = (a, b, c) \) must be perpendicular to both \( \vec{n_1} \) and \( \vec{n_2} \). This gives us two equations:
1. \( 2a + b - 5c = 0 \) (from \( \vec{n} \cdot \vec{n_1} = 0 \))
2. \( 3a + 5b - 7c = 0 \) (from \( \vec{n} \cdot \vec{n_2} = 0 \))
### Step 3: Solve the system of equations
From the first equation:
\[
b = -2a + 5c
\]
Substituting \( b \) into the second equation:
\[
3a + 5(-2a + 5c) - 7c = 0
\]
\[
3a - 10a + 25c - 7c = 0
\]
\[
-7a + 18c = 0 \implies 7a = 18c \implies a = \frac{18}{7}c
\]
Since \( a, b, c \) must be integers, let \( c = 7k \) for some integer \( k \):
\[
a = 18k, \quad b = -2(18k) + 5(7k) = -36k + 35k = -k
\]
Thus, we have:
\[
a = 18k, \quad b = -k, \quad c = 7k
\]
### Step 4: Use the point (2, 3, -5) to find \( d \)
Substituting the point \( (2, 3, -5) \) into the plane equation:
\[
18k(2) - k(3) + 7k(-5) = d
\]
\[
36k - 3k - 35k = d \implies -2k = d
\]
Since \( d > 0 \), we have \( k < 0 \). Let \( k = -m \) where \( m \) is a positive integer:
\[
a = 18(-m) = -18m, \quad b = -(-m) = m, \quad c = 7(-m) = -7m, \quad d = 2m
\]
### Step 5: Find the GCD condition
We need \( \gcd(|a|, |b|, |c|, d) = 1 \):
\[
\gcd(18m, m, 7m, 2m) = m \cdot \gcd(18, 1, 7, 2) = m
\]
To satisfy the condition \( \gcd = 1 \), we must have \( m = 1 \).
### Step 6: Substitute \( m = 1 \)
Thus, we have:
\[
a = -18, \quad b = 1, \quad c = -7, \quad d = 2
\]
### Step 7: Calculate \( a + 7b + c + 20d \)
Now we can calculate:
\[
a + 7b + c + 20d = -18 + 7(1) - 7 + 20(2)
\]
\[
= -18 + 7 - 7 + 40 = -18 + 40 = 22
\]
### Final Answer
The value of \( a + 7b + c + 20d \) is \( \boxed{22} \).
