In order that the number 1 y 3 y 6 be divisible by 11, the digit y should be

Options

A. 1
B. 2
C. 5
D. 6

Answer

To determine the value of \( y \) such that the number \( 1y3y6 \) is divisible by 11, we can use the divisibility rule for 11. According to this rule, for a number to be divisible by 11, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions must either be 0 or a multiple of 11.

### Step-by-step Solution:

1. **Identify the positions of the digits**:
The number \( 1y3y6 \) has the following positions:
- Odd positions: 1 (1st), 3 (3rd), 6 (5th)
- Even positions: \( y \) (2nd), \( y \) (4th)

2. **Calculate the sum of the digits in odd positions**:
\[ \text{Sum of odd positions} = 1 + 3 + 6 = 10 \]

3. **Calculate the sum of the digits in even positions**:
\[ \text{Sum of even positions} = y + y = 2y \]

4. **Set up the equation based on the divisibility rule**:
According to the rule, we need:
\[ |(10) - (2y)| \equiv 0 \mod 11 \]
This means \( 10 - 2y \) should be either 0 or a multiple of 11.

5. **Solve for \( y \)**:
We can set up the equation:
\[ 10 - 2y = 0 \implies 2y = 10 \implies y = 5 \]

6. **Check if this value satisfies the divisibility**:
Substitute \( y = 5 \) back into the number:
The number becomes \( 15356 \).
- Odd positions: \( 1 + 3 + 6 = 10 \)
- Even positions: \( 5 + 5 = 10 \)
- Difference: \( |10 - 10| = 0 \), which is divisible by 11.

Thus, the digit \( y \) should be \( 5 \).

### Final Answer:
The value of \( y \) is \( 5 \).