To draw the Hasse diagram of the poset \((D_{625}, \div)\), we first need to understand what this notation means:

- \(D_{625}\) is the set of all divisors of 625.
- The relation \(\div\) represents divisibility.

### Steps to Construct the Hasse Diagram:

1. **List all divisors of 625**: Since \(625 = 5^4\), its divisors are:
   \[
   D_{625} = \{1, 5, 25, 125, 625\}
   \]
   
2. **Order the elements using the divisibility relation**: In a Hasse diagram, we only draw an edge between two elements \(a\) and \(b\) if \(a \div b\) and there is no \(c\) such that \(a \div c \div b\).

3. **Determine the relations**:
   - \(1 \div 5, 25, 125, 625\)
   - \(5 \div 25, 125, 625\)
   - \(25 \div 125, 625\)
   - \(125 \div 625\)

4. **Draw the Hasse diagram**:
   - Place \(1\) at the bottom.
   - \(5\) is directly above \(1\).
   - \(25\) is above \(5\).
   - \(125\) is above \(25\).
   - \(625\) is at the top.

This structure is a straight line because each divisor is divisible by exactly one other divisor.

### Visual Representation:

```
   625
    |
   125
    |
   25
    |
    5
    |
    1
```

This vertical line represents the Hasse diagram of \((D_{625}, \div)\), where each element is connected to the one below it by divisibility.

Would you like any additional details or explanations? Here are some questions to explore further:

1. What is a Hasse diagram, and how is it used in other contexts?
2. How do we determine the divisors of a number like 625?
3. Can Hasse diagrams be non-linear, and in what situations?
4. How does the divisibility relation influence the structure of the Hasse diagram?
5. What are some other applications of posets in mathematics?

**Tip:** In Hasse diagrams, if you can't draw a direct connection between two elements without passing through others, it's not necessary to include the direct connection, as the diagram reflects the minimal relationships that maintain the partial order.

Learn how to construct the Hasse diagram of the poset (D625, /), where D625 represents the divisors of 625. This problem covers key concepts in order theory, including posets, divisibility, and the structure of Hasse diagrams. Suitable for undergraduate mathematics students.

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