We need to find a set of integers of the form $3qrs14$ where:

• $3 < q < r < s < 14$ (i.e., $q, r, s$ are distinct integers between 4 and 13)
• The PDL (Prime Divisor List) contains no repeated integers

Step 1: Understanding the Prime Divisor List (PDL)

The Prime Divisor List (PDL) of a number is the list of all distinct prime factors of that number.

For $3qrs14$, we first determine its prime factorization:

$3qrs14 = 3qrs \times 14 = 3qrs \times (2 \times 7)$

Thus, the PDL of $3qrs14$ is:

$\text{PDL} = \{2, 3, 7\} \cup \text{PDL of } (qrs)$

To ensure no repeated integers in the PDL, $q, r, s$ must be chosen such that their prime factorizations introduce no duplicates from $\{2,3,7\}$ or among themselves.

Step 2: Choosing $q, r, s$

We select values from $4 \leq q < r < s \leq 13$, ensuring they introduce only new prime factors.

Example Set:

Let’s choose $q = 5$, $r = 9$, and $s = 11$.

1. Prime factorizations:
   • $5$ (prime)
   • $9 = 3^2$ (introduces $3$, which is already in PDL)
   • $11$ (prime)

Since $3$ is already in the PDL due to 3 itself, including $9$ would repeat $3$, which we must avoid.

Let’s try another set: $q = 5, r = 8, s = 11$.

1. Prime factorizations:
   • $5$ (prime)
   • $8 = 2^3$ (introduces $2$, already in PDL)
   • $11$ (prime)

Again, $2$ is repeated.

A better set is:
$q = 5, r = 9, s = 13$

1. Prime factorizations:
   • $5$ (prime)
   • $9 = 3^2$ (introduces $3$, already in PDL)
   • $13$ (prime)

Since 9 introduces a repeated $3$, we need to remove it.

A valid choice is $q = 5, r = 10, s = 11$.

1. Prime factorizations:
   • $5$ (prime)
   • $10 = 2 \times 5$ (introduces $2, 5$)
   • $11$ (prime)

Thus, the PDL is:

$\{2, 3, 5, 7, 11\}$

which contains no repeated numbers.

Answer:

A valid set is $\{3, 5, 10, 11, 14\}$.

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Do you need more examples or a deeper breakdown? Here are five related questions to think about:

1. How can we systematically find all valid sets satisfying the conditions?
2. What happens if we allow repeated integers in the PDL?
3. How does the PDL change if we replace 14 with another number?
4. What is the effect of choosing composite numbers for $q, r, s$?
5. Can we construct a general formula for sets that satisfy this condition?

Tip: When constructing sets with distinct prime factor lists, start by ensuring new primes are introduced at each step!