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Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Source: Caucasus MO 2023

July 16, 2023

number theory

Problem Statement

Let

n{}

and

m

be positive integers,

n>m>1

. Let

n{}

divided by

m

have partial quotient

q

and remainder

r

(so that

n = qm + r

, where

r\in\{0,1,...,m-1\}

). Let

n-1

divided by

m

have partial quotient

q^{'}

and remainder

r^{'}

. a) It appears that

q+q^{'} =r +r^{'} = 99

. Find all possible values of

n{}

. b) Prove that if

q+q^{'} =r +r^{'}

, then

2n

is a perfect square.

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