If a nine-digit number 785x3678y is divisible by 72, then the value of (x + y) is:

Options

A. 5
B. 12
C. 10
D. 20

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the nine-digit number \( 785x3678y \) such that it is divisible by 72. A number is divisible by 72 if it is divisible by both 8 and 9.

### Step 1: Check divisibility by 8
For a number to be divisible by 8, the last three digits of the number must be divisible by 8. The last three digits in our case are \( 78y \).

To find \( y \), we can check the values of \( y \) from 0 to 9 and see which makes \( 78y \) divisible by 8.

- \( 780 \div 8 = 97.5 \) (not divisible)
- \( 781 \div 8 = 97.625 \) (not divisible)
- \( 782 \div 8 = 97.75 \) (not divisible)
- \( 783 \div 8 = 97.875 \) (not divisible)
- \( 784 \div 8 = 98 \) (divisible)
- \( 785 \div 8 = 98.125 \) (not divisible)
- \( 786 \div 8 = 98.25 \) (not divisible)
- \( 787 \div 8 = 98.375 \) (not divisible)
- \( 788 \div 8 = 98.5 \) (not divisible)
- \( 789 \div 8 = 98.625 \) (not divisible)

Thus, the only value of \( y \) that makes \( 78y \) divisible by 8 is \( y = 4 \).

### Step 2: Check divisibility by 9
Now we need to ensure that the entire number \( 785x36784 \) is divisible by 9. For a number to be divisible by 9, the sum of its digits must be divisible by 9.

Calculating the sum of the digits:

\[ 7 + 8 + 5 + x + 3 + 6 + 7 + 8 + 4 = 48 + x \]

We need \( 48 + x \) to be divisible by 9.

Calculating \( 48 \mod 9 \):

\[ 48 \div 9 = 5 \quad \text{(remainder 3)} \]

So, \( 48 \equiv 3 \mod 9 \).

To make \( 48 + x \equiv 0 \mod 9 \), we need:

\[ 3 + x \equiv 0 \mod 9 \implies x \equiv 6 \mod 9 \]

The possible values for \( x \) that satisfy this condition and are single digits are \( x = 6 \).

### Step 3: Calculate \( x + y \)
Now that we have \( x = 6 \) and \( y = 4 \):

\[ x + y = 6 + 4 = 10 \]

### Final Answer
Thus, the value of \( x + y \) is \( \boxed{10} \).