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q^2 + r = 2011 when ab = q (a + b) + r , 0 \le r <a + b

Source: Germany Federal - Bundeswettbewerb Mathematik 2011, round 1, p4

April 10, 2020

Diophantine equationdiophantineEuclidean algorithmnumber theory

Problem Statement

Let

a

and

b

be positive integers. As is known, the division of of

a \cdot b

with

a + b

determines integers

q

and

r

uniquely such that

a \cdot b = q (a + b) + r

and

0 \le r <a + b

. Find all pairs

(a, b)

for which

q^2 + r = 2011