Understanding Divisibility by 8

The question asks us to identify which number among a given set is not divisible by 8. To solve this, we need to use the divisibility rule for 8.

The divisibility rule for 8 states that a whole number is divisible by 8 if the number formed by its last three digits is divisible by 8. If the number has fewer than three digits, we check if the number itself is divisible by 8.

Applying the Divisibility Rule for 8

Let's examine each number provided:

• 98984: The last three digits form the number 984. We need to check if 984 is divisible by 8.
• 98992: The last three digits form the number 992. We need to check if 992 is divisible by 8.
• 98998: The last three digits form the number 998. We need to check if 998 is divisible by 8.
• 99008: The last three digits form the number 008, which is simply 8. We need to check if 8 is divisible by 8.

Step-by-Step Division Check

We will now divide the number formed by the last three digits of each given number by 8 to see if the remainder is zero.

1. For 98984:

   Check 984:

   \[ \frac{984}{8} \]

   Performing the division:

   \[ 984 \div 8 \]

   8 goes into 9 once (remainder 1). Bring down 8, making 18. 8 goes into 18 twice (16, remainder 2). Bring down 4, making 24. 8 goes into 24 three times (24, remainder 0).

   \[ 984 \div 8 = 123 \]

   Since 984 is divisible by 8, the number 98984 is divisible by 8.

2. For 98992:

   Check 992:

   \[ \frac{992}{8} \]

   Performing the division:

   \[ 992 \div 8 \]

   8 goes into 9 once (remainder 1). Bring down 9, making 19. 8 goes into 19 twice (16, remainder 3). Bring down 2, making 32. 8 goes into 32 four times (32, remainder 0).

   \[ 992 \div 8 = 124 \]

   Since 992 is divisible by 8, the number 98992 is divisible by 8.

3. For 98998:

   Check 998:

   \[ \frac{998}{8} \]

   Performing the division:

   \[ 998 \div 8 \]

   8 goes into 9 once (remainder 1). Bring down 9, making 19. 8 goes into 19 twice (16, remainder 3). Bring down 8, making 38. 8 goes into 38 four times (32, remainder 6).

   \[ 998 \div 8 = 124 \text{ with a remainder of } 6 \]

   Since 998 is not divisible by 8 (it has a remainder), the number 98998 is NOT divisible by 8.

4. For 99008:

   Check 008 (or 8):

   \[ \frac{8}{8} \]

   Performing the division:

   \[ 8 \div 8 = 1 \]

   Since 8 is divisible by 8, the number 99008 is divisible by 8.

Summary of Divisibility Checks

Based on the checks, the number 98998 is the only one among the given options whose last three digits (998) are not divisible by 8. Therefore, 98998 is not divisible by 8.

Revision Table: Divisibility Rules

Additional Information: Why the Divisibility Rule for 8 Works

The divisibility rule for 8 is based on the fact that \(1000\) is divisible by 8 (\(1000 \div 8 = 125\)). Any number can be written as \(1000 \times (\text{some number}) + (\text{last three digits})\). For example, the number 98984 can be written as \(98 \times 1000 + 984\).

If a number \(N\) is written as \(N = 1000 \times Q + R\), where \(Q\) is the quotient and \(R\) is the remainder when \(N\) is divided by 1000 (so \(R\) is the number formed by the last three digits), then \(N\) is divisible by 8 if and only if \(1000 \times Q + R\) is divisible by 8.

Since \(1000\) is divisible by 8, \(1000 \times Q\) is always divisible by 8, regardless of the value of \(Q\). Therefore, for \(N\) to be divisible by 8, the remainder \(R\) (the number formed by the last three digits) must also be divisible by 8. This explains why we only need to check the last three digits.