If `(pqr)^2 = (ijk pqr),` where `i,j,k,p,q,r in W,` are pqr and ijkpqr are three digits and 6 digits numbers respectively. Then the value of `i xx j xx k xx p xx q xx r ` is :

Options

A. 0
B. 720
C. can't say
D. None of these

Answer

To solve the equation \((pqr)^2 = (ijkpqr)\), where \(pqr\) is a three-digit number and \(ijkpqr\) is a six-digit number, we need to follow these steps:

### Step 1: Understand the Range of \(pqr\)
Since \(pqr\) is a three-digit number, it must be in the range of 100 to 999.

### Step 2: Determine the Range of \((pqr)^2\)
Next, we calculate the square of the smallest and largest three-digit numbers:
- The smallest three-digit number is \(100\):
\[ (100)^2 = 10000 \]
- The largest three-digit number is \(999\):
\[ (999)^2 = 998001 \]
Thus, \((pqr)^2\) must be a six-digit number, which means it must be between \(100000\) and \(999999\).

### Step 3: Set the Condition for \(ijkpqr\)
The six-digit number \(ijkpqr\) can be expressed as:
\[ ijkpqr = 1000 \times ijk + pqr \]
This means that \((pqr)^2\) must equal \(1000 \times ijk + pqr\).

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ (pqr)^2 - pqr = 1000 \times ijk \]
Factoring out \(pqr\) on the left side:
\[ pqr(pqr - 1) = 1000 \times ijk \]

### Step 5: Analyzing the Equation
Since \(1000 \times ijk\) is divisible by \(1000\), \(pqr(pqr - 1)\) must also be divisible by \(1000\). This means that \(pqr\) must be such that the product \(pqr(pqr - 1)\) is a multiple of \(1000\).

### Step 6: Finding Suitable Values for \(pqr\)
To satisfy the divisibility condition, \(pqr\) must end in \(0\) (so that \(pqr\) is divisible by \(10\)) or \(pqr - 1\) must end in \(0\) (so that \(pqr - 1\) is divisible by \(10\)).

### Step 7: Testing Values
Let's test some values of \(pqr\):
1. If \(pqr = 100\):
\[ (100)^2 = 10000 \quad \text{(not a six-digit number)} \]
2. If \(pqr = 316\):
\[ (316)^2 = 99856 \quad \text{(not a six-digit number)} \]
3. If \(pqr = 320\):
\[ (320)^2 = 102400 \quad \text{(valid)} \]
Here, \(ijk = 102\) and \(pqr = 400\).

### Step 8: Calculate \(i \times j \times k \times p \times q \times r\)
From \(ijk = 102\) and \(pqr = 400\):
- \(i = 1\), \(j = 0\), \(k = 2\), \(p = 4\), \(q = 0\), \(r = 0\).

Calculating the product:
\[ i \times j \times k \times p \times q \times r = 1 \times 0 \times 2 \times 4 \times 0 \times 0 = 0 \]

### Final Answer
The value of \(i \times j \times k \times p \times q \times r\) is \(0\).