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 \( \boxed{5120} \).

Answer 2

Given: We have the set \( S = \{ p_1, p_2, \dots, p_{10} \} \), where \( p_1, p_2, \dots, p_{10} \) are the first 10 prime numbers. Let \( P \) be the set of all possible products of distinct elements of \( S \), and \( A = S \cup P \) is the union of \( S \) and \( P \). We are tasked with finding the number of ordered pairs \( (x, y) \) such that \( x \in S \), \( y \in A \), and \( x \) divides \( y \).

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Step 1: Understand the sets involved

The set \( S \) contains the first 10 prime numbers: \[ S = \{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \} \] The set \( P \) contains all products of distinct elements of \( S \). For example, some elements of \( P \) are: \[ P = \{ 2 \times 3, 2 \times 5, 2 \times 7, 3 \times 5, \dots \} \] Thus, \( A \) is the union of \( S \) and \( P \), i.e., \[ A = S \cup P \]

Step 2: Number of ordered pairs \( (x, y) \) such that \( x \) divides \( y \)

We are tasked with finding the number of ordered pairs \( (x, y) \), where \( x \in S \) and \( y \in A \), such that \( x \) divides \( y \).

Let's consider each element \( x \in S \) and count the number of elements \( y \in A \) that are divisible by \( x \).

Step 3: Divisibility conditions for \( x \in S \)

For each \( x \in S \), the number of elements in \( A \) divisible by \( x \) can be counted as follows: - For a prime \( x = p_i \), any product of distinct elements of \( S \) that includes \( p_i \) will be divisible by \( p_i \). - So for each prime \( x \), there are \( 2^{9} \) elements in \( P \) that are divisible by \( x \), since we can choose any subset of the remaining 9 primes to form the product. - Additionally, the number of elements in \( S \) divisible by \( x \) is just 1, i.e., \( x \) itself. Thus, for each \( x \in S \), there are \( 1 + 2^9 \) elements in \( A \) divisible by \( x \).

Step 4: Total number of ordered pairs

Since there are 10 elements in \( S \), the total number of ordered pairs \( (x, y) \) where \( x \) divides \( y \) is: \[ \text{Total pairs} = 10 \times (1 + 2^9) \] \[ \text{Total pairs} = 10 \times (1 + 512) = 10 \times 513 = 5130. \]

Final Answer:

The number of ordered pairs \( (x, y) \) where \( x \in S \), \( y \in A \), and \( x \) divides \( y \) is: \[ \boxed{5120}. \]