2y5 is divisible by 11, find the value of digit y.

Answer

To find the value of the digit \( y \) in the number \( 2y5 \) such that it is divisible by 11, we can follow these steps:

### Step 1: Understand the divisibility rule for 11
The rule states that a number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or divisible by 11.

### Step 2: Identify the positions of the digits in \( 2y5 \)
In the number \( 2y5 \):
- The digits in the odd positions are: \( 2 \) (1st position) and \( 5 \) (3rd position).
- The digit in the even position is: \( y \) (2nd position).

### Step 3: Calculate the sums
- Sum of the digits in odd positions: \( 2 + 5 = 7 \)
- Sum of the digits in the even position: \( y \)

### Step 4: Set up the equation for divisibility by 11
According to the rule:
\[ \text{Difference} = (\text{Sum of odd position digits}) - (\text{Sum of even position digits}) = 7 - y \]
For \( 2y5 \) to be divisible by 11, \( 7 - y \) must be either 0 or divisible by 11.

### Step 5: Solve for \( y \)
We can set up the equation:
1. \( 7 - y = 0 \)
- This gives \( y = 7 \)

2. \( 7 - y = 11k \) for some integer \( k \)
- For \( k = 1 \): \( 7 - y = 11 \) → \( y = -4 \) (not valid)
- For \( k = -1 \): \( 7 - y = -11 \) → \( y = 18 \) (not valid)
- Higher values of \( k \) will also yield invalid digits (greater than 9 or negative).

### Step 6: Conclusion
The only valid solution for \( y \) is:
\[ y = 7 \]
Thus, the digit \( y \) that makes \( 2y5 \) divisible by 11 is \( 7 \).

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