If a nine-digit number 785x3678y is divisible by 72, then the value of `(7x-5y)` is:

Options

A. 22
B. 20
C. 14
D. 25

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) such that the nine-digit number \( 785x3678y \) 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
To check if the number is divisible by 8, we only need to consider the last three digits, which are \( 78y \).

**Hint:** A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

We need to find \( y \) such that \( 78y \) is divisible by 8. We can test values for \( y \) from 0 to 9:

- \( 780 \div 8 = 97.5 \) (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
Next, we check the divisibility by 9. A number is divisible by 9 if the sum of its digits is divisible by 9.

The digits of the number are \( 7, 8, 5, x, 3, 6, 7, 8, 4 \). The sum of these digits is:

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

**Hint:** A number is divisible by 9 if the sum of its digits is divisible by 9.

Now, we need \( 48 + x \) to be divisible by 9. We can find \( 48 \mod 9 \):

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

So, \( 48 \equiv 3 \mod 9 \). For \( 48 + x \) to be divisible by 9, \( x \) must satisfy:

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

The possible values for \( x \) are \( 6 \) (since \( x \) must be a single digit).

### Step 3: Calculate \( 7x - 5y \)
Now that we have \( x = 6 \) and \( y = 4 \), we can calculate \( 7x - 5y \):

\[ 7x - 5y = 7(6) - 5(4) = 42 - 20 = 22 \]

### Final Answer
Thus, the value of \( 7x - 5y \) is \( \boxed{22} \).