The digits of a three digit number are in G.P. when the digits of this number are reversed and this resultant number is subtracted from the original number the difference comes out to be 792. The actual number is:

Options

A. 842
B. 961
C. 421
D. 931

Answer

To solve the problem step by step, we will analyze the conditions given in the question and find the three-digit number whose digits are in a geometric progression (G.P.) and meets the difference condition when reversed.

### Step 1: Understanding the problem
We need to find a three-digit number \(abc\) where \(a\), \(b\), and \(c\) are the digits of the number. The digits must be in G.P., and when the number is reversed to \(cba\), the difference between the original number and the reversed number should be 792.

### Step 2: Setting up the equations
Let the three-digit number be represented as:
\[ N = 100a + 10b + c \]
The reversed number will be:
\[ R = 100c + 10b + a \]
According to the problem, we have:
\[ N - R = 792 \]
Substituting the expressions for \(N\) and \(R\):
\[ (100a + 10b + c) - (100c + 10b + a) = 792 \]
Simplifying this gives:
\[ 99a - 99c = 792 \]
Dividing the entire equation by 99:
\[ a - c = 8 \quad \text{(1)} \]

### Step 3: Finding the G.P. condition
The digits \(a\), \(b\), and \(c\) are in G.P., which means:
\[ \frac{b}{a} = \frac{c}{b} \implies b^2 = ac \quad \text{(2)} \]

### Step 4: Using the equations
From equation (1), we can express \(c\) in terms of \(a\):
\[ c = a - 8 \]
Substituting \(c\) into equation (2):
\[ b^2 = a(a - 8) \]
This simplifies to:
\[ b^2 = a^2 - 8a \quad \text{(3)} \]

### Step 5: Finding possible values for \(a\), \(b\), and \(c\)
Since \(a\), \(b\), and \(c\) are digits (0-9), we need to find integer solutions for equations (1) and (3).

1. From \(c = a - 8\), \(a\) must be at least 8 (to keep \(c\) a digit).
2. Therefore, \(a\) can be 8 or 9.

#### Case 1: \(a = 9\)
\[ c = 9 - 8 = 1 \]
Substituting into equation (3):
\[ b^2 = 9(1) = 9 \implies b = 3 \]
So, one possible number is \(931\).

#### Case 2: \(a = 8\)
\[ c = 8 - 8 = 0 \]
Substituting into equation (3):
\[ b^2 = 8(0) = 0 \implies b = 0 \]
So, another possible number is \(800\), but it doesn't satisfy the G.P. condition.

### Step 6: Verifying the solution
Now we check the number \(931\):
- The digits are \(9\), \(3\), and \(1\).
- They are in G.P. because \(3^2 = 9 \times 1\).

Now, check the difference:
\[ 931 - 139 = 792 \]
This satisfies the condition.

### Final Answer
The actual number is:
\[ \boxed{931} \]