If 34N is divisible by 11, then what is the value of N?

Options

A. a) 1
B. b) 3
C. c) 4
D. d) 9

Answer

To find the value of \( N \) such that \( 34N \) is divisible by 11, we can use the divisibility rule for 11. Here’s a step-by-step solution:

### Step 1: Identify the digits in the number
The number \( 34N \) consists of the digits 3, 4, and \( N \).

### Step 2: Apply the divisibility rule for 11
According to the divisibility rule for 11, we need to find the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions.

- Odd positions: 3 (1st position) and \( N \) (3rd position)
- Even positions: 4 (2nd position)

### Step 3: Calculate the sums
- Sum of digits in odd positions: \( 3 + N \)
- Sum of digits in even positions: \( 4 \)

### Step 4: Set up the equation
According to the rule, the difference should be a multiple of 11:
\[ (3 + N) - 4 = 0 \quad \text{(or a multiple of 11)} \]
This simplifies to:
\[ N - 1 = 0 \]

### Step 5: Solve for \( N \)
From the equation \( N - 1 = 0 \), we find:
\[ N = 1 \]

### Conclusion
Thus, the value of \( N \) that makes \( 34N \) divisible by 11 is \( N = 1 \).

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