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Rational number can be written uniquely as sum of a_i/i!

Source: 2017 Thailand October Camp 1.1

February 25, 2022

number theoryfactorial

Problem Statement

Prove that any rational

r \in (0, 1)

can be written uniquely in the form

r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}

where

a_i\text{’s}

are nonnegative integers with

a_i\leq i-1

for all

i