Let n be a positive integer and Dn
be the set of all positive divisor of n then
Dn
is a lattice under the relation of divisibility and find the compliment of
every element of lattice D20 = {1,2,4,5,10,20},D30 = {1,2,3,5,6,10,15,30}

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.

Let n be a positive integer and Dn be the set of all positive divisors of n. Then Dn is a lattice under the relation of divisibility. Find the complement of every element of the lattices D20 = {1, 2, 4, 5, 10, 20} and D30 = {1, 2, 3, 5, 6, 10, 15, 30}.

Explore how to find the complements of elements in the divisor lattices D20 = {1, 2, 4, 5, 10, 20} and D30 = {1, 2, 3, 5, 6, 10, 15, 30}. Learn key concepts in divisibility, greatest common divisor (GCD), and least common multiple (LCM). Suitable for advanced high school and university-level students.

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