If an eleven-digit number `5y5888406xx6` is divisible by 72, then what is the value of (9x — 2y), for the least value of x?

Options

A. 5
B. 3
C. 4
D. 7

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the eleven-digit number \( 5y5888406xx6 \) such that the number 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 need to look at the last three digits of the number, which are \( xx6 \).

1. The last three digits can be expressed as \( 100x + 10x + 6 = 110x + 6 \).
2. We need to find values of \( x \) such that \( 110x + 6 \) is divisible by 8.

Now, let's check for the least value of \( x \):

- If \( x = 0 \): \( 110(0) + 6 = 6 \) (not divisible by 8)
- If \( x = 1 \): \( 110(1) + 6 = 116 \) (divisible by 8)
- If \( x = 2 \): \( 110(2) + 6 = 226 \) (not divisible by 8)
- If \( x = 3 \): \( 110(3) + 6 = 336 \) (not divisible by 8)
- If \( x = 4 \): \( 110(4) + 6 = 446 \) (not divisible by 8)
- If \( x = 5 \): \( 110(5) + 6 = 556 \) (not divisible by 8)
- If \( x = 6 \): \( 110(6) + 6 = 666 \) (not divisible by 8)
- If \( x = 7 \): \( 110(7) + 6 = 776 \) (divisible by 8)
- If \( x = 8 \): \( 110(8) + 6 = 886 \) (not divisible by 8)
- If \( x = 9 \): \( 110(9) + 6 = 996 \) (not divisible by 8)

The least value of \( x \) that makes \( xx6 \) divisible by 8 is \( x = 1 \).

### Step 2: Check divisibility by 9
Next, we need to check if the entire number is divisible by 9. For this, we sum all the digits of the number \( 5 + y + 5 + 8 + 8 + 8 + 4 + 0 + 6 + x + 6 \).

1. The sum of the digits is \( 5 + y + 5 + 8 + 8 + 8 + 4 + 0 + 6 + 1 + 6 = 41 + y \).
2. We need \( 41 + y \) to be divisible by 9.

Now, let's find suitable values for \( y \):

- If \( y = 0 \): \( 41 + 0 = 41 \) (not divisible by 9)
- If \( y = 1 \): \( 41 + 1 = 42 \) (not divisible by 9)
- If \( y = 2 \): \( 41 + 2 = 43 \) (not divisible by 9)
- If \( y = 3 \): \( 41 + 3 = 44 \) (not divisible by 9)
- If \( y = 4 \): \( 41 + 4 = 45 \) (divisible by 9)
- If \( y = 5 \): \( 41 + 5 = 46 \) (not divisible by 9)
- If \( y = 6 \): \( 41 + 6 = 47 \) (not divisible by 9)
- If \( y = 7 \): \( 41 + 7 = 48 \) (divisible by 9)
- If \( y = 8 \): \( 41 + 8 = 49 \) (not divisible by 9)
- If \( y = 9 \): \( 41 + 9 = 50 \) (not divisible by 9)

The possible values of \( y \) that make \( 41 + y \) divisible by 9 are \( y = 4 \) and \( y = 7 \). However, we need to find the least value of \( x \), which we already found to be \( x = 1 \).

### Step 3: Calculate \( 9x - 2y \)
Now, we can calculate \( 9x - 2y \) using \( x = 1 \) and \( y = 4 \):

\[ 9x - 2y = 9(1) - 2(4) = 9 - 8 = 1 \]

### Final Answer
The value of \( 9x - 2y \) for the least value of \( x \) is **1**.