If x48y is exactly divisible by 9, then the least value of (x+y) is

Options

A. 4
B. 0
C. 6
D. 7

Answer

To determine the least value of \( x + y \) such that the number \( x48y \) is exactly divisible by 9, we can follow these steps:

### Step 1: Understand the divisibility rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

### Step 2: Identify the digits of the number
The digits of the number \( x48y \) are \( x, 4, 8, y \).

### Step 3: Calculate the sum of the known digits
The sum of the known digits is:
\[ 4 + 8 = 12 \]
Thus, the total sum of the digits becomes:
\[ x + 12 + y \]

### Step 4: Set up the divisibility condition
For \( x48y \) to be divisible by 9, the sum \( x + 12 + y \) must be divisible by 9.

### Step 5: Express the condition mathematically
Let \( S = x + y + 12 \). We need \( S \equiv 0 \mod 9 \).

### Step 6: Find the smallest possible value of \( S \)
The smallest multiple of 9 that is greater than or equal to 12 is 18. Therefore, we set:
\[ x + y + 12 = 18 \]
This simplifies to:
\[ x + y = 18 - 12 = 6 \]

### Step 7: Determine the values of \( x \) and \( y \)
Now we need to find non-negative integer values for \( x \) and \( y \) such that:
\[ x + y = 6 \]
The pairs \( (x, y) \) that satisfy this equation are:
- \( (0, 6) \)
- \( (1, 5) \)
- \( (2, 4) \)
- \( (3, 3) \)
- \( (4, 2) \)
- \( (5, 1) \)
- \( (6, 0) \)

### Step 8: Find the least value of \( x + y \)
The least value of \( x + y \) that satisfies the condition is \( 6 \).

### Conclusion
Thus, the least value of \( x + y \) is \( 6 \).

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