If 4xy7 is exactly divisible by 3, then the least value of (x+y) is

Options

A. 1
B. 4
C. 5
D. 7

Answer

To find the least value of \( x + y \) such that the number \( 4xy7 \) is exactly divisible by 3, we can 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: Write down the digits of the number
The digits of the number \( 4xy7 \) are \( 4, x, y, \) and \( 7 \).

### Step 3: Calculate the sum of the known digits
Add the known digits:
\[ 4 + 7 = 11 \]
So, the total sum of the digits becomes:
\[ 11 + x + y \]

### Step 4: Set up the condition for divisibility by 3
For \( 4xy7 \) to be divisible by 3, the sum \( 11 + x + y \) must be divisible by 3.

### Step 5: Find the remainder of 11 when divided by 3
Calculate \( 11 \mod 3 \):
\[ 11 \div 3 = 3 \quad \text{(remainder 2)} \]
Thus, \( 11 \equiv 2 \mod 3 \).

### Step 6: Set up the equation for \( x + y \)
For \( 11 + x + y \equiv 0 \mod 3 \):
\[ 2 + x + y \equiv 0 \mod 3 \]
This simplifies to:
\[ x + y \equiv 1 \mod 3 \]

### Step 7: Find possible values of \( x + y \)
The smallest non-negative values of \( x + y \) that satisfy \( x + y \equiv 1 \mod 3 \) are:
- \( x + y = 1 \)
- \( x + y = 4 \)
- \( x + y = 7 \)
- etc.

### Step 8: Determine the least value of \( x + y \)
The least value of \( x + y \) that satisfies the condition is:
\[ x + y = 1 \]

### Conclusion
Thus, the least value of \( x + y \) is \( 1 \).

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