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if \frac{a+1}{b}+ \frac{b+1}{a} is integer then gcd(a,b)<=\sqrt{a+b}

Source: Spanish Mathematical Olympiad 1996 P1

July 31, 2018

number theorygreatest common divisorInteger

Problem Statement

The natural numbers

a

and

b

are such that

\frac{a+1}{b}+ \frac{b+1}{a}

is an integer. Show that the greatest common divisor of a and b is not greater than

\sqrt{a+b}