A Multiplicative Closure Property

★ 8/10Geometric Progression number theory Set Theory algebra sequences

Problem

A set $A = \{a_1, \dots, a_n\}$ with $1 \le a_1 < a_2 < \cdots < a_n$ and $n \ge 2$ has property P: for all $1 \le i \le j \le n$ , at least one of $a_i a_j$ and $\dfrac{a_j}{a_i}$ belongs to $A$ .

1. Do $\{1, 3, 4\}$ and $\{1, 2, 3, 6\}$ have property P? 2. Prove that $a_1 = 1$ and

\frac{a_1 + a_2 + \cdots + a_n}{a_1^{-1} + a_2^{-1} + \cdots + a_n^{-1}} = a_n.

Prove that for $n = 5$ the five elements form a geometric progression.

Answer

Solution

Difficulty	8/10
Topics	Geometric Progression, number theory, Set Theory, algebra, sequences

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