If R019 is divisible by 11, find the value of the smallest natural number R ?
A. 5
B. 6
C. 7
D. 8

Options

A. B
B. A
C. C
D. D

Answer

To determine the smallest natural number \( R \) such that the number \( R019 \) 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 a multiple of 11.

### Step 2: Identify the positions of the digits in \( R019 \)
The number \( R019 \) can be broken down as follows:
- Odd positions: \( R \) (1st position) and \( 0 \) (3rd position)
- Even positions: \( 0 \) (2nd position) and \( 1 \) (4th position)

### Step 3: Calculate the sums
- Sum of digits at odd positions: \( R + 0 = R \)
- Sum of digits at even positions: \( 0 + 1 = 1 \)

### Step 4: Find the difference
Now, we calculate the difference:
\[ \text{Difference} = \text{Sum of odd positions} - \text{Sum of even positions} = R - 1 \]

### Step 5: Set up the condition for divisibility by 11
For \( R019 \) to be divisible by 11, the difference \( R - 1 \) must be 0 or a multiple of 11:
\[ R - 1 = 0 \quad \text{or} \quad R - 1 = 11k \quad (k \text{ is an integer}) \]

### Step 6: Solve for \( R \)
1. Setting \( R - 1 = 0 \):
\[ R = 1 \]

2. Setting \( R - 1 = 11 \):
\[ R = 12 \]
(not a valid single digit)

### Step 7: Check for multiples of 11
The next multiple of 11 after 0 is 11, which gives:
\[ R - 1 = 11 \implies R = 12 \quad (\text{not valid as } R \text{ must be a single digit}) \]

### Step 8: Check the smallest natural number
Since \( R \) must be a natural number and a single digit, we check the values from 1 to 9:
- For \( R = 1 \): \( 1 - 1 = 0 \) (valid)
- For \( R = 2 \): \( 2 - 1 = 1 \) (not valid)
- For \( R = 3 \): \( 3 - 1 = 2 \) (not valid)
- For \( R = 4 \): \( 4 - 1 = 3 \) (not valid)
- For \( R = 5 \): \( 5 - 1 = 4 \) (not valid)
- For \( R = 6 \): \( 6 - 1 = 5 \) (not valid)
- For \( R = 7 \): \( 7 - 1 = 6 \) (not valid)
- For \( R = 8 \): \( 8 - 1 = 7 \) (not valid)
- For \( R = 9 \): \( 9 - 1 = 8 \) (not valid)

### Conclusion
The smallest natural number \( R \) that makes \( R019 \) divisible by 11 is:
\[ \boxed{1} \]