If 21y5 is a multiple of 9, where y is a digit, what is the value of y?

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To determine the value of \( y \) in the number \( 21y5 \) such that it is a multiple of 9, we will use the divisibility rule for 9. According to this rule, a number is divisible by 9 if the sum of its digits is divisible by 9.

### Step-by-Step Solution:

1. **Identify the digits of the number**: The number is \( 21y5 \), where the digits are 2, 1, \( y \), and 5.

2. **Calculate the sum of the known digits**:
\[ 2 + 1 + 5 = 8 \]

3. **Include \( y \) in the sum**:
\[ \text{Total sum} = 8 + y \]

4. **Set up the condition for divisibility by 9**:
We need \( 8 + y \) to be divisible by 9. This means:
\[ 8 + y \equiv 0 \ (\text{mod} \ 9) \]

5. **Find possible values of \( y \)**:
To find \( y \), we can check which digit (from 0 to 9) makes \( 8 + y \) divisible by 9. We can calculate:
- If \( y = 0 \): \( 8 + 0 = 8 \) (not divisible by 9)
- If \( y = 1 \): \( 8 + 1 = 9 \) (divisible by 9)
- If \( y = 2 \): \( 8 + 2 = 10 \) (not divisible by 9)
- If \( y = 3 \): \( 8 + 3 = 11 \) (not divisible by 9)
- If \( y = 4 \): \( 8 + 4 = 12 \) (not divisible by 9)
- If \( y = 5 \): \( 8 + 5 = 13 \) (not divisible by 9)
- If \( y = 6 \): \( 8 + 6 = 14 \) (not divisible by 9)
- If \( y = 7 \): \( 8 + 7 = 15 \) (not divisible by 9)
- If \( y = 8 \): \( 8 + 8 = 16 \) (not divisible by 9)
- If \( y = 9 \): \( 8 + 9 = 17 \) (not divisible by 9)

6. **Conclusion**: The only value of \( y \) that makes \( 21y5 \) a multiple of 9 is \( y = 1 \).

### Final Answer:
The value of \( y \) is \( 1 \).

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