A 5 - digit number xy235 is divisible by 3 such that ` x + y lt 5` where x and y are single digits then possible values of (x, y) are _____

Options

A. (1,1) or (4, 0)
B. (1,1) or (2, 0)
C. (1,1) or (0,3)
D. (2, 0) or (0,2)

Answer

To solve the problem step-by-step, we need to find the values of \( x \) and \( y \) such that the 5-digit number \( xy235 \) is divisible by 3 and \( x + y < 5 \).

### Step 1: Understand the divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

### Step 2: Calculate the sum of the known digits
The known digits in the number \( xy235 \) are 2, 3, and 5. We can calculate their sum:
\[ 2 + 3 + 5 = 10 \]

### Step 3: Include \( x \) and \( y \) in the sum
Now, we add \( x \) and \( y \) to this sum:
\[ \text{Total sum} = x + y + 10 \]

### Step 4: Set up the divisibility condition
For \( xy235 \) to be divisible by 3, \( x + y + 10 \) must be divisible by 3. We can express this as:
\[ x + y + 10 \equiv 0 \, (\text{mod} \, 3) \]

### Step 5: Simplify the condition
Since \( 10 \mod 3 = 1 \), we can rewrite the condition:
\[ x + y + 1 \equiv 0 \, (\text{mod} \, 3) \]
This simplifies to:
\[ x + y \equiv 2 \, (\text{mod} \, 3) \]

### Step 6: Consider the constraint \( x + y < 5 \)
We also have the condition that \( x + y < 5 \). Therefore, the possible values for \( x + y \) that satisfy both conditions are:
- \( x + y = 2 \)
- \( x + y = 5 \) is not valid since it does not satisfy \( x + y < 5 \).

### Step 7: Find pairs \((x, y)\) that satisfy \( x + y = 2 \)
The pairs of non-negative single-digit integers \((x, y)\) that satisfy \( x + y = 2 \) are:
- \( (0, 2) \)
- \( (1, 1) \)
- \( (2, 0) \)

### Step 8: Verify the pairs against the condition \( x + y < 5 \)
All the pairs \( (0, 2) \), \( (1, 1) \), and \( (2, 0) \) satisfy \( x + y < 5 \).

### Final Answer
Thus, the possible values of \( (x, y) \) are:
- \( (0, 2) \)
- \( (1, 1) \)
- \( (2, 0) \)