If 31y5 is divisible by 15, where y is a single digit, then what would be the values of y?

Options

A. 1, 2, 5,8
B. 0, 3, 6, 9
C. 2, 4, 6, 8
D. 4, 3, 8, 7

Answer

To determine the values of \( y \) such that the number \( 31y5 \) is divisible by 15, we need to check the conditions for divisibility by both 3 and 5.

### Step-by-Step Solution:

1. **Check for divisibility by 5**:
- A number is divisible by 5 if its last digit is either 0 or 5.
- Here, the last digit of \( 31y5 \) is 5, which is already divisible by 5.
- Therefore, \( 31y5 \) satisfies the condition for divisibility by 5.

**Hint**: Remember that for a number to be divisible by 5, it must end in 0 or 5.

2. **Check for divisibility by 3**:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- The digits of \( 31y5 \) are 3, 1, \( y \), and 5.
- Calculate the sum of the digits:
\[ 3 + 1 + y + 5 = 9 + y \]

**Hint**: To check divisibility by 3, sum all the digits and see if that sum is divisible by 3.

3. **Determine when \( 9 + y \) is divisible by 3**:
- Since 9 is already divisible by 3, \( 9 + y \) will be divisible by 3 if \( y \) itself is divisible by 3.
- The single-digit numbers that are divisible by 3 are 0, 3, 6, and 9.

**Hint**: List the single-digit numbers (0-9) and check which of them are divisible by 3.

4. **Possible values for \( y \)**:
- The values of \( y \) that satisfy the condition \( y \equiv 0 \mod 3 \) are:
- \( y = 0 \)
- \( y = 3 \)
- \( y = 6 \)
- \( y = 9 \)

### Conclusion:
The possible values of \( y \) such that \( 31y5 \) is divisible by 15 are:
- \( y = 0 \)
- \( y = 3 \)
- \( y = 6 \)
- \( y = 9 \)