In each of the following questions , three statements of numbers following same rules are given . Find the rule and accordingly find the value of the number ?
If ` 2 xx 1 = 81 , 3 xx 2 = 278 , 2 xx 5 = 8125` , then `1 xx 3 = ?`

Options

A. 127
B. 271
C. 126
D. 129

Explanation

To solve the problem step by step, we need to identify the rule used in the given statements and apply it to find the value of `1 xx 3`.

### Step 1: Analyze the given statements
We have the following statements:
1. \( 2 \times 1 = 81 \)
2. \( 3 \times 2 = 278 \)
3. \( 2 \times 5 = 8125 \)

### Step 2: Identify the pattern
Let's break down the first statement:
- \( 2 \times 1 = 81 \)
- Here, \( 2^3 = 8 \) and \( 1^3 = 1 \)
- Combining these gives us \( 81 \) (which is \( 8 \) followed by \( 1 \)).

Now, let's check the second statement:
- \( 3 \times 2 = 278 \)
- \( 3^3 = 27 \) and \( 2^3 = 8 \)
- Combining these gives us \( 278 \) (which is \( 27 \) followed by \( 8 \)).

Next, we check the third statement:
- \( 2 \times 5 = 8125 \)
- \( 2^3 = 8 \) and \( 5^3 = 125 \)
- Combining these gives us \( 8125 \) (which is \( 8 \) followed by \( 125 \)).

### Step 3: Formulate the rule
From the above analysis, we can conclude the rule:
For \( a \times b \), the result is formed by concatenating \( a^3 \) and \( b^3 \).

### Step 4: Apply the rule to find \( 1 \times 3 \)
Now we need to find \( 1 \times 3 \):
- \( 1^3 = 1 \)
- \( 3^3 = 27 \)

Combining these results:
- The result is \( 1 \) followed by \( 27 \), which gives us \( 127 \).

### Final Answer
Thus, \( 1 \times 3 = 127 \).

---