`3x2` divisible by 11 ?

Options

A. `5`
B. `6`
C. `7`
D. `8`

Answer

To determine if the expression \(3x2\) is divisible by 11, we will follow these steps:

### Step 1: Understand the expression
The expression \(3x2\) represents a three-digit number where \(x\) is a digit. The number can be broken down as follows:
- The hundreds place is 3
- The tens place is \(x\)
- The units place is 2

### Step 2: Identify the odd and even positioned digits
In the number \(3x2\):
- Odd positioned digits: 3 (hundreds place) and 2 (units place)
- Even positioned digit: \(x\) (tens place)

### Step 3: Calculate the sum of the odd and even positioned digits
- Sum of odd positioned digits: \(3 + 2 = 5\)
- Sum of even positioned digits: \(x\)

### Step 4: Find the difference
To check for divisibility by 11, we need to find the difference between the sum of the odd positioned digits and the sum of the even positioned digits:
\[ \text{Difference} = \text{Sum of odd digits} - \text{Sum of even digits} = 5 - x \]

### Step 5: Check for divisibility by 11
For \(3x2\) to be divisible by 11, the difference \(5 - x\) must be divisible by 11. We will check the possible values of \(x\) (which can be any digit from 0 to 9).

### Step 6: Test values of \(x\)
We will substitute values from 0 to 9 for \(x\) and check if \(5 - x\) is divisible by 11:
- If \(x = 0\): \(5 - 0 = 5\) (not divisible)
- If \(x = 1\): \(5 - 1 = 4\) (not divisible)
- If \(x = 2\): \(5 - 2 = 3\) (not divisible)
- If \(x = 3\): \(5 - 3 = 2\) (not divisible)
- If \(x = 4\): \(5 - 4 = 1\) (not divisible)
- If \(x = 5\): \(5 - 5 = 0\) (divisible by 11)
- If \(x = 6\): \(5 - 6 = -1\) (not divisible)
- If \(x = 7\): \(5 - 7 = -2\) (not divisible)
- If \(x = 8\): \(5 - 8 = -3\) (not divisible)
- If \(x = 9\): \(5 - 9 = -4\) (not divisible)

### Conclusion
The only value of \(x\) that makes \(3x2\) divisible by 11 is \(x = 5\). Therefore, the answer is:
\[ \text{The value of } x \text{ is } 5. \]

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