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: Understanding the set \( A \)

The set \( A = S \cup P \) consists of \( S \), the set of the first ten primes, and \( P \), the set of all possible products of distinct elements of \( S \). Thus, \( |A| = 2^{10} - 1 = 1023 \), since there are \( 2^{10} \) subsets of \( S \), excluding the empty subset.

Step 2: Counting the pairs \( (x, y) \) where \( x \) divides \( y \)

For each \( x \in S \), \( x \) divides exactly half of the elements of \( A \), as for every product that doesn't contain \( x \), there is a corresponding one that does. Hence, for each \( x \in S \), there are 512 elements in \( A \) divisible by \( x \).

Step 3: Total number of pairs

Since there are 10 elements in \( S \), the total number of ordered pairs \( (x, y) \) such that \( x \) divides \( y \) is:

\[ 512 \times 10 = 5120. \]

Final Answer:

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