MathDB

1/r is an integer when r min of (a/b - c/d) when gcd (a, b) = 1, c<=a, d <= b

Source: 2020 Dutch IMO TST 1.4

November 21, 2020

number theoryInteger

Problem Statement

Let

a, b \ge 2

be positive integers with

gcd (a, b) = 1

. Let

r

be the smallest positive value that

\frac{a}{b}- \frac{c}{d}

can take, where

c

and

d

are positive integers satisfying

c \le a

and

d \le b

. Prove that

\frac{1}{r}

is an integer.