In the given expression `pq = p - q + 9, q` is a fraction and p is any positive integer. The value of p, which is inadmissible is :

Options

A. 5
B. 4
C. 8
D. 7

Answer

To solve the problem, we start with the given expression:

**Step 1: Write down the equation.**
We have the equation:
\[ pq = p - q + 9 \]

**Step 2: Rearrange the equation.**
We can rearrange this equation to isolate \( q \):
\[ pq + q = p + 9 \]
This can be factored as:
\[ q(p + 1) = p + 9 \]

**Step 3: Solve for \( q \).**
Now, we can solve for \( q \):
\[ q = \frac{p + 9}{p + 1} \]

**Step 4: Identify the condition for \( q \).**
Since \( q \) is a fraction, \( p + 9 \) must not be divisible by \( p + 1 \). This means that \( p + 9 \) should not be a multiple of \( p + 1 \).

**Step 5: Analyze the divisibility condition.**
We can express the condition mathematically:
\[ p + 9 = k(p + 1) \]
for some integer \( k \). Rearranging gives:
\[ p + 9 = kp + k \]
This leads to:
\[ p(1 - k) = k - 9 \]

**Step 6: Determine inadmissible values of \( p \).**
To find the values of \( p \) that make \( q \) an integer, we can substitute different positive integer values for \( p \) and check if \( q \) remains a fraction.

- For \( p = 4 \):
\[ q = \frac{4 + 9}{4 + 1} = \frac{13}{5} \] (fraction)

- For \( p = 5 \):
\[ q = \frac{5 + 9}{5 + 1} = \frac{14}{6} = \frac{7}{3} \] (fraction)

- For \( p = 6 \):
\[ q = \frac{6 + 9}{6 + 1} = \frac{15}{7} \] (fraction)

- For \( p = 7 \):
\[ q = \frac{7 + 9}{7 + 1} = \frac{16}{8} = 2 \] (integer)

- For \( p = 8 \):
\[ q = \frac{8 + 9}{8 + 1} = \frac{17}{9} \] (fraction)

From the above calculations, we see that when \( p = 7 \), \( q \) is an integer, which is inadmissible.

**Final Answer:**
The value of \( p \) which is inadmissible is \( 7 \).

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