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upper bound of quotient

Source: INMO 2014- Problem 3

February 2, 2014

number theorygreatest common divisorleast common multipleinequalitiesnumber theory proposed

Problem Statement

Let

a,b

be natural numbers with

ab>2

. Suppose that the sum of their greatest common divisor and least common multiple is divisble by

a+b

. Prove that the quotient is at most

\frac{a+b}{4}

. When is this quotient exactly equal to

\frac{a+b}{4}