Find all possible values of y for which the number 53y1 is divisible by 3. Also , find each such number.

Answer

To find all possible values of \( y \) for which the number \( 53y1 \) is divisible by 3, we need to follow these steps:

### 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: Identify the digits of the number
The number we are considering is \( 53y1 \). The digits are:
- 5
- 3
- \( y \)
- 1

### Step 3: Calculate the sum of the known digits
First, we calculate the sum of the known digits:
\[ 5 + 3 + 1 = 9 \]

### Step 4: Include \( y \) in the sum
Now, we include \( y \) in the sum:
\[ 9 + y \]

### Step 5: Set up the condition for divisibility by 3
For \( 53y1 \) to be divisible by 3, the sum \( 9 + y \) must be divisible by 3. Since 9 is already divisible by 3, \( y \) must also be such that \( y \) itself is divisible by 3.

### Step 6: Determine possible values of \( y \)
Since \( y \) is a single-digit number, the possible values of \( y \) that are divisible by 3 are:
- 0
- 3
- 6
- 9

### Step 7: List the numbers formed by each possible value of \( y \)
Now we can substitute each possible value of \( y \) into the number \( 53y1 \):
- If \( y = 0 \), the number is \( 5301 \)
- If \( y = 3 \), the number is \( 5331 \)
- If \( y = 6 \), the number is \( 5361 \)
- If \( y = 9 \), the number is \( 5391 \)

### Final Answer
The possible values of \( y \) are \( 0, 3, 6, \) and \( 9 \). The corresponding numbers are:
- \( 5301 \)
- \( 5331 \)
- \( 5361 \)
- \( 5391 \)