Question

The sum of all possible values of \(x\) so that the 9 digit number \(384x579x\) is divisible by \(6\) is

• A. \(6\)
• B. \(8\)
• C. \(10\)
• D. \(12\)

Hint:

For divisibility by \(6\), always check divisibility by \(2\) and \(3\) separately.

Answer 1

The Correct Option is A

Concept: A number is divisible by \(6\) if it is divisible by both: \[ 2 \text{ and } 3 \]

Step 1: Check divisibility by 2.
The last digit is \(x\). For divisibility by \(2\): \[ x \text{ must be even} \] So possible values: \[ 0,2,4,6,8 \]

Step 2: Check divisibility by 3.
Sum of digits: \[ 3+8+4+x+5+7+9+x \] \[ =36+2x \] For divisibility by \(3\): \[ 36+2x \] must be divisible by \(3\). Since \(36\) is divisible by \(3\), \[ 2x \text{ must be divisible by } 3 \] Thus: \[ x \text{ must be divisible by } 3 \] Possible values: \[ 0,3,6,9 \]

Step 3: Find common values.
Even values: \[ 0,2,4,6,8 \] Multiples of 3: \[ 0,3,6,9 \] Common: \[ 0,6 \]

Step 4: Find their sum.
\[ 0+6=6 \] Thus, the required answer is: \[ \boxed{6} \]