1. **Problem statement:** Determine all the sub-lattices of $D_{30} = \{1,2,3,5,6,10,15,30\}$ that contain at least four elements. 2. **Background:** $D_{30}$ is the set of divisors of 30, ordered by divisibility. A sub-lattice is a subset closed under the meet (greatest common divisor, gcd) and join (least common multiple, lcm) operations. 3. **Key properties:** - The meet of two elements $a,b$ is $\gcd(a,b)$. - The join of two elements $a,b$ is $\mathrm{lcm}(a,b)$. - A sub-lattice must contain the meet and join of any two of its elements. 4. **Step-by-step approach:** - Identify subsets of $D_{30}$ with at least 4 elements. - Check closure under gcd and lcm for each candidate subset. 5. **Check the full lattice:** $D_{30}$ itself has 8 elements and is a lattice. 6. **Check sub-lattices with 4 or more elements:** - Consider subsets containing 1 and 30 (minimum and maximum) for lattice completeness. - Example 1: $\{1,2,6,30\}$ - gcd and lcm of any two elements remain in the set. - Check $\gcd(2,6)=2$, $\mathrm{lcm}(2,6)=6$ both in set. - Check $\gcd(6,30)=6$, $\mathrm{lcm}(6,30)=30$ both in set. - Check $\gcd(1,2)=1$, $\mathrm{lcm}(1,2)=2$ both in set. - So this is a sub-lattice. - Example 2: $\{1,3,15,30\}$ - Similar checks show closure. - Example 3: $\{1,5,10,30\}$ - Also closed under gcd and lcm. - Example 4: $\{1,2,3,6,30\}$ - Check closure: - $\gcd(2,3)=1$, $\mathrm{lcm}(2,3)=6$ in set. - $\gcd(3,6)=3$, $\mathrm{lcm}(3,6)=6$ in set. - $\gcd(6,30)=6$, $\mathrm{lcm}(6,30)=30$ in set. - Closed. - Example 5: $\{1,2,5,10,30\}$ - Check closure: - $\gcd(2,5)=1$, $\mathrm{lcm}(2,5)=10$ in set. - $\gcd(5,10)=5$, $\mathrm{lcm}(5,10)=10$ in set. - $\gcd(10,30)=10$, $\mathrm{lcm}(10,30)=30$ in set. - Closed. - Example 6: $\{1,3,5,15,30\}$ - Check closure: - $\gcd(3,5)=1$, $\mathrm{lcm}(3,5)=15$ in set. - $\gcd(5,15)=5$, $\mathrm{lcm}(5,15)=15$ in set. - $\gcd(15,30)=15$, $\mathrm{lcm}(15,30)=30$ in set. - Closed. 7. **Summary of sub-lattices with at least 4 elements:** - $\{1,2,6,30\}$ - $\{1,3,15,30\}$ - $\{1,5,10,30\}$ - $\{1,2,3,6,30\}$ - $\{1,2,5,10,30\}$ - $\{1,3,5,15,30\}$ - $D_{30}$ itself These are all sub-lattices of $D_{30}$ with at least four elements. **Final answer:** The sub-lattices of $D_{30}$ with at least four elements are exactly the sets listed above.