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Let

u

be a positive rational number and

m

be a positive integer. Define a sequence

q_1,q_2,q_3,\dotsc

such that

q_1=u

and for

n\geqslant 2

\text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}.

Determine all positive integers

m

such that the sequence

q_1,q_2,q_3,\dotsc

is eventually periodic for any positive rational number

u

.
Remark: A sequence

x_1,x_2,x_3,\dotsc

is eventually periodic if there are positive integers

c

and

t

such that

x_n=x_{n+t}

for all

n\geqslant c

.
Proposed by Petar Nizié-Nikolac

Problem Statement