Question

Let \( S = \{p_1, p_2, \dots, p_{10}\} \) be the set of the first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y) \), where \( x \in S \), \( y \in A \), and \( x \) divides \( y \), is _________.

Hint:

When working with divisibility problems involving sets of primes, remember that the number of elements divisible by a particular prime is half of the total number of subsets, excluding the empty subset.

Answer 1

Step 1: Define Set A

Set \( A = S \cup P \) comprises \( S \), the first ten prime numbers, and \( P \), all products formed by distinct elements of \( S \). The cardinality of \( A \) is \( |A| = 2^{10} - 1 = 1023 \), as \( 2^{10} \) represents the total number of subsets of \( S \), excluding the empty set.

Step 2: Count Divisible Pairs per Element in S

For every element \( x \in S \), it divides precisely half of the elements in \( A \). This is because for every product in \( A \) that does not include \( x \), there exists a corresponding product that does. Consequently, for each \( x \in S \), there are 512 elements in \( A \) that are divisible by \( x \).

Step 3: Calculate Total Divisible Pairs

Given that \( S \) contains 10 elements, the total count of ordered pairs \( (x, y) \) where \( x \) divides \( y \) is:

\[ 512 \times 10 = 5120. \]

Final Answer:

The final answer is \( \boxed{5120} \).