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Ratios of terms of arithmetic progressions

Source: 44th International Tournament of Towns, Senior O-Level P4, Spring 2023

January 9, 2024

number theoryArithmetic Progression

Problem Statement

Let

a_1, a_2, a_3,\ldots

and

b_1, b_2, b_3,\ldots

be infinite increasing arithmetic progressions. Their terms are positive numbers. It is known that the ratio

a_k/b_k

is an integer for all k. Is it true that this ratio does not depend on

k{}

?
Boris Frenkin