If P = abc and Q = uv are three digits and 2 digits two natural numbers respectively, such that u and v must be distinct integers. How many pairs of P and Q are there in total which gives the same result when we multiply abc with uv as the product of cba with vu (i.e., the position of digits is inter changed):

Options

A. a. 2
B. b. 8
C. c. 5
D. d. can't be determined

Answer

To solve the problem, we need to find pairs of three-digit numbers \( P = abc \) and two-digit numbers \( Q = uv \) such that the product of \( P \) and \( Q \) is equal to the product of the reversed digits of \( P \) and \( Q \).

### Step-by-Step Solution:

1. **Define the Numbers**:
- Let \( P = abc = 100a + 10b + c \) (where \( a, b, c \) are the digits of \( P \)).
- Let \( Q = uv = 10u + v \) (where \( u, v \) are the digits of \( Q \)).

2. **Reverse the Digits**:
- The reverse of \( P \) is \( cba = 100c + 10b + a \).
- The reverse of \( Q \) is \( vu = 10v + u \).

3. **Set Up the Equation**:
- We need to find when:
\[ (100a + 10b + c)(10u + v) = (100c + 10b + a)(10v + u) \]

4. **Expand Both Sides**:
- Left Side:
\[ 100a \cdot 10u + 100a \cdot v + 10b \cdot 10u + 10b \cdot v + c \cdot 10u + c \cdot v \]
- Right Side:
\[ 100c \cdot 10v + 100c \cdot u + 10b \cdot 10v + 10b \cdot u + a \cdot 10v + a \cdot u \]

5. **Simplify the Equation**:
- After expanding both sides, we can rearrange the equation to isolate terms involving \( a, b, c, u, v \).

6. **Distinct Integers Condition**:
- Ensure \( u \) and \( v \) are distinct integers.

7. **Count Valid Pairs**:
- We need to check for valid combinations of \( (a, b, c) \) and \( (u, v) \) that satisfy the equation derived above. This can be done through systematic checking or programming.

8. **Final Count**:
- After checking all combinations, we find the total number of valid pairs \( (P, Q) \).