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We are given two equations in the problem:

1. \( 2x + 5y - 150 = 0 \)
2. \( kx + 5y = 10 \)

We are asked to determine the value of \( b \) when \( (8, b) \) is a solution of the second equation.

### Step 1: Solve for \( y \) in terms of \( x \)
First, let's solve the first equation for possible solutions of \( x \) and \( y \) (whole numbers):

\[
2x + 5y = 150
\]

Rearrange for \( 5y \):

\[
5y = 150 - 2x
\]

Now, divide both sides by 5:

\[
y = \frac{150 - 2x}{5}
\]

For \( y \) to be a whole number, the expression \( 150 - 2x \) must be divisible by 5. Therefore, \( 2x \equiv 150 \pmod{5} \), which simplifies to:

\[
2x \equiv 0 \pmod{5}
\]

This means \( x \) must be a multiple of 5. Let \( x = 5n \), where \( n \) is an integer. Substituting this into the equation for \( y \):

\[
y = \frac{150 - 2(5n)}{5} = \frac{150 - 10n}{5} = 30 - 2n
\]

Thus, the number of possible solutions for \( x \) and \( y \) is based on the values \( n \) can take. Since \( x \) must be non-negative, \( 0 \leq x \leq 75 \), meaning \( 0 \leq n \leq 15 \). So, there are 16 possible solutions.

### Step 2: Solve for \( b \) when \( (8, b) \) is a solution
Now, substitute \( x = 8 \) into the second equation \( kx + 5y = 10 \):

\[
k(8) + 5b = 10
\]

This simplifies to:

\[
8k + 5b = 10
\]

Solve for \( b \):

\[
5b = 10 - 8k
\]

\[
b = \frac{10 - 8k}{5}
\]

For \( b \) to be a whole number, \( 10 - 8k \) must be divisible by 5. This gives us the condition:

\[
10 - 8k \equiv 0 \pmod{5}
\]

Simplifying:

\[
8k \equiv 10 \pmod{5}
\]

\[
3k \equiv 0 \pmod{5}
\]

Since 3 and 5 are relatively prime, \( k \) must be a multiple of 5. Let \( k = 5m \), where \( m \) is an integer. Substituting this into the equation for \( b \):

\[
b = \frac{10 - 8(5m)}{5} = \frac{10 - 40m}{5} = 2 - 8m
\]

### Step 3: Determine \( b \) for different values of \( k \)
For different values of \( m \):

- If \( m = 0 \), \( b = 2 \)
- If \( m = 1 \), \( b = -6 \)
- If \( m = -1 \), \( b = 10 \)

Now, from the options given, none of these values directly match. Therefore, it's possible the equation has been transformed or given a fraction option we missed earlier. Upon solving

The number of possible solutions of the equation 2x + 5y - 150 = 0, where x and y are whole numbers is k. The value of b if (8, b) is a solution of kx + 5y = 10 is:

This problem involves solving a system of linear equations with whole number solutions. We also find the value of 'b' for a specific value of 'x'. Learn how to apply divisibility rules and modular arithmetic to solve for multiple variables.

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