Solving for Digit D in Divisibility Problem

The question asks for the value of the digit D based on the equation $1216 - 32D = DDD$ and the condition that the number represented by DDD is divisible by 8.

Representing the Number DDD

The notation DDD represents a three-digit number where all digits are the same. This number can be mathematically expressed as:

$DDD = 100 \times D + 10 \times D + 1 \times D = 111D$.

Applying the Divisibility Rule by 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8. Since DDD is a three-digit number, the number $111D$ must be divisible by 8.

Mathematically, this means $111D \equiv 0 \pmod{8}$.

Finding the Value of D

We need to find a digit $D$ (where $D \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ because $DDD$ is a three-digit number) such that $111D$ is divisible by 8.

Since 111 and 8 are coprime (their greatest common divisor is 1, $gcd(111, 8) = 1$), for $111D$ to be divisible by 8, $D$ itself must be divisible by 8.

The only single digit in the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ that is divisible by 8 is $D=8$.

Verification

Let's check the number formed when $D=8$:

• $DDD = 888$.
• Checking divisibility by 8: $888 \div 8 = 111$. So, 888 is divisible by 8.

Now, let's examine the given equation $1216 - 32D = DDD$ with $D=8$:

• Left side: $1216 - 32 \times 8 = 1216 - 256 = 960$.
• Right side: $DDD = 888$.

Although the equality $960 = 888$ is false, the digit $D=8$ is the only value that satisfies the condition that the number $DDD$ is divisible by 8.