State whether each of the following sets is a finite set or an infinite set:
The set of multiples of 8

Answer

To determine whether the set of multiples of 8 is a finite set or an infinite set, we can follow these steps:

### Step 1: Define the Set of Multiples of 8
The set of multiples of 8 can be expressed as:
\[ S = \{ 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, \ldots \} \]
This set includes all numbers that can be obtained by multiplying 8 by any integer.

### Step 2: Identify the Pattern
The multiples of 8 can be represented mathematically as:
\[ S = \{ 8n \mid n \in \mathbb{Z} \} \]
where \( n \) is any integer (0, 1, 2, 3, ...).

### Step 3: Determine Countability
Since \( n \) can take any integer value (positive, negative, or zero), there is no limit to how many multiples of 8 can be generated. For example:
- For \( n = 0 \): \( 8 \times 0 = 0 \)
- For \( n = 1 \): \( 8 \times 1 = 8 \)
- For \( n = 2 \): \( 8 \times 2 = 16 \)
- For \( n = 3 \): \( 8 \times 3 = 24 \)
- This continues indefinitely.

### Step 4: Conclusion
Since there is no upper limit to the values of \( n \) and the set continues indefinitely, we conclude that the set of multiples of 8 is an **infinite set**.

### Final Answer
The set of multiples of 8 is an infinite set.

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