We are tasked with finding the complement of each element in the divisor lattices $D_{20}$ and $D_{30}$, where the lattices are based on the divisibility relation.

Key Concepts:

1. Divisor Lattice: A set of divisors of a number forms a lattice under divisibility. In this lattice, for two elements $a$ and $b$, the meet (greatest lower bound) is the greatest common divisor (gcd), and the join (least upper bound) is the least common multiple (lcm).

2. Complement in a Lattice: For an element $x$ in a lattice, the complement $x^c$ is defined (in some contexts) as an element such that the meet of $x$ and $x^c$ is the minimal element (1 in this case), and the join is the maximal element (which is $n$ in this case, where $n = 20$ or $n = 30$).

Thus, for a divisor $d$ of $n$, its complement $d^c$ is the divisor that "pairs" with it in the sense that: $\text{gcd}(d, d^c) = 1 \quad \text{and} \quad \text{lcm}(d, d^c) = n.$ This means that $d^c = \frac{n}{d}$.

Step-by-Step Solution:

Let’s now apply this to $D_{20}$ and $D_{30}$.

For $D_{20} = \{ 1, 2, 4, 5, 10, 20 \}$:

1. The complement of 1: $1^c = \frac{20}{1} = 20$
2. The complement of 2: $2^c = \frac{20}{2} = 10$
3. The complement of 4: $4^c = \frac{20}{4} = 5$
4. The complement of 5: $5^c = \frac{20}{5} = 4$
5. The complement of 10: $10^c = \frac{20}{10} = 2$
6. The complement of 20: $20^c = \frac{20}{20} = 1$

Thus, the complements in $D_{20}$ are: $\{ (1, 20), (2, 10), (4, 5), (5, 4), (10, 2), (20, 1) \}.$

For $D_{30} = \{ 1, 2, 3, 5, 6, 10, 15, 30 \}$:

1. The complement of 1: $1^c = \frac{30}{1} = 30$
2. The complement of 2: $2^c = \frac{30}{2} = 15$
3. The complement of 3: $3^c = \frac{30}{3} = 10$
4. The complement of 5: $5^c = \frac{30}{5} = 6$
5. The complement of 6: $6^c = \frac{30}{6} = 5$
6. The complement of 10: $10^c = \frac{30}{10} = 3$
7. The complement of 15: $15^c = \frac{30}{15} = 2$
8. The complement of 30: $30^c = \frac{30}{30} = 1$

Thus, the complements in $D_{30}$ are: $\{ (1, 30), (2, 15), (3, 10), (5, 6), (6, 5), (10, 3), (15, 2), (30, 1) \}.$

Conclusion:

In both divisor lattices $D_{20}$ and $D_{30}$, each element has a complement that satisfies the conditions $\text{gcd}(d, d^c) = 1$ and $\text{lcm}(d, d^c) = n$.

Would you like more details on how to visualize these lattices or further clarification?

Follow-Up Questions:

1. How are gcd and lcm used to find meets and joins in divisor lattices?
2. Can you explain how lattices are structured in terms of divisibility?
3. What are some practical applications of divisor lattices?
4. How would the complements change if we used a different $n$, like $n = 12$?
5. How can we verify that a pair of elements are complements in a lattice?

Tip:

Understanding how gcd and lcm relate to lattice operations (meet and join) is key to analyzing divisibility lattices effectively.