Olympiad Maths Prep

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Problem 691

AIME late
Number theory Difficulty 5.3 Prove it

25.41. Prove that the number ee is irrational.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

25.41. Suppose that e=m/ne=m / n, where mm and nn are natural numbers. Then according to problem 25.39

0<mn−(2+12!+…+1n!)<1n!n 0<\frac{m}{n}-\left(2+\frac{1}{2!}+\ldots+\frac{1}{n!}\right)<\frac{1}{n!n}

After multiplying by n!n! we get that 0<a<1/n0<a<1 / n, where aa is an integer. This cannot be

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.