Hashtag

We have sets A n A_{n} and B n B_{n} , n > 1 n>1

Set A n A_{n} contains all natural numbers more than 1 that are divisible by some prime p p that satisfies p n p\mid n .

Set B n B_{n} contains all natural numbers more than 1 that are divisible by some prime p p that satisfies p ∤ n p \not\mid n .

Define N N = set of integers greater than 1

Which of the following is true?

A n B n = N A_n \cup B_n\ = N and A n , B n A_n, B_n are not disjoint A n B n = N A_n \cup B_n\ = N and A n , B n A_n, B_n are disjoint A n B n N A_n \cup B_n\ \not= N None of the rest

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1 solution

Julian Yu
Dec 3, 2018

Every integer greater than 1 1 is divisible by a prime - call this prime p p . Since either p n p\mid n or p ∤ n p\not\mid n , N N is the union of A n A_n and B n B_n .

Let p p be a prime that divides n n and q q be a prime that doesn't divide n n . The number p q pq is an element of both A n A_n and B n B_n , so the two sets are not disjoint.

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