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Square root of 11....44....4

Source: Romanian District Olympiad 2006, Grade 7, Problem 1

March 11, 2006
modular arithmetic

Problem Statement

Prove that for all positive integers nn, n>1n>1 the number 11…44…4‾\sqrt{ \overline{ 11\ldots 44 \ldots 4 }}, where 1 appears nn times, and 4 appears 2n2n times, is irrational.
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