Let $S=\{p_1,p_2,\dots,p_{10}\}$ 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 ______.

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] For a fixed prime $x\in S$, the elements of $A$ divisible by $x$ are the products of distinct primes that include $x$ (a single prime counts as the product of one element): choose any subset of the other 9 primes, giving $2^9=512$. Total $=10	imes512=5120$. Answer: 5120.

Q. Let $S=\{p_1,p_2,\dots,p_{10}\}$ 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 ______.

Explanation

Reviewed by Dr. Abubacker

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