Let be the set of first ten prime numbers, Let , where P is the set of all possible products of distinct elements of S. Then the number of all ordered pairs , such that x divides y, is __________. [2025]

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Solution

Q.
Let $S = {p_{1}, p_{2}, . . . . . ., p_{10}}$ be the set of 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 $(x, y), x \in S, y \in A$ , such that x divides y, is __________. [2025]

Ans.
(5120)

S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

$P = {2 \times 3, 2 \times 3 \times 5, . . .}$

$A = S \cup P = {2, 2 \times 3, . . ., 3; 3 \times 5, . . .}$

For x = 2, the value of y can be

$1 + {}^{9}C_{1} + {}^{9}C_{2} + {}^{9}C_{3} + . . . + {}^{9}C_{9} = 2^{9}$

Similarly, for x = 3, 5, 7, 11, ...; y can be $2^{9}$

$∴$ Required number of ordered pair = $10 \times (2^{9})$

= $10 \times 512 = 5120$

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