What is the value of x so that the seven digit number 55350 x 2 is divisible by 72 ?

Options

A. 1
B. 8
C. 7
D. 3

Answer

To find the value of \( x \) such that the seven-digit number \( 55350x2 \) is divisible by 72, we need to ensure that it is divisible by both 8 and 9, since \( 72 = 8 \times 9 \).

### 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 of our number are \( 0x2 \).

We will check the values of \( x \) from 0 to 9 to see which makes \( 0x2 \) divisible by 8.

- **If \( x = 0 \)**: Last three digits = 002 → \( 2 \div 8 = 0.25 \) (not divisible)
- **If \( x = 1 \)**: Last three digits = 012 → \( 12 \div 8 = 1.5 \) (not divisible)
- **If \( x = 2 \)**: Last three digits = 022 → \( 22 \div 8 = 2.75 \) (not divisible)
- **If \( x = 3 \)**: Last three digits = 032 → \( 32 \div 8 = 4 \) (divisible)
- **If \( x = 4 \)**: Last three digits = 042 → \( 42 \div 8 = 5.25 \) (not divisible)
- **If \( x = 5 \)**: Last three digits = 052 → \( 52 \div 8 = 6.5 \) (not divisible)
- **If \( x = 6 \)**: Last three digits = 062 → \( 62 \div 8 = 7.75 \) (not divisible)
- **If \( x = 7 \)**: Last three digits = 072 → \( 72 \div 8 = 9 \) (divisible)
- **If \( x = 8 \)**: Last three digits = 082 → \( 82 \div 8 = 10.25 \) (not divisible)
- **If \( x = 9 \)**: Last three digits = 092 → \( 92 \div 8 = 11.5 \) (not divisible)

From this check, \( x = 3 \) and \( x = 7 \) make the number divisible by 8.

### Step 2: Check divisibility by 9
Next, we need to check if the sum of the digits of \( 55350x2 \) is divisible by 9. The sum of the digits is \( 5 + 5 + 3 + 5 + 0 + x + 2 = 15 + x \).

- **If \( x = 3 \)**: Sum = \( 15 + 3 = 18 \) → \( 18 \div 9 = 2 \) (divisible)
- **If \( x = 7 \)**: Sum = \( 15 + 7 = 22 \) → \( 22 \div 9 \approx 2.44 \) (not divisible)

### Conclusion
The only value of \( x \) that makes \( 55350x2 \) divisible by both 8 and 9 is \( x = 3 \).

Thus, the value of \( x \) is **3**.

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