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VTRMC
2011 VTRMC
Problem 4
Problem 4
Part of
2011 VTRMC
Problems
(1)
residue classes, existence where [a] is class mod mn
Source: VTRMC 2011 P4
5/14/2021
Let
m
,
n
m,n
m
,
n
be positive integers and let
[
a
]
[a]
[
a
]
denote the residue class
(
m
o
d
m
n
)
\pmod{mn}
(
mod
mn
)
of the integer
a
a
a
(thus
{
[
r
]
∣
r
is an integer
}
\{[r]|r\text{ is an integer}\}
{[
r
]
∣
r
is an integer
}
has exactly
m
n
mn
mn
elements). Suppose the set
{
[
a
r
]
∣
r
is an integer
}
\{[ar]|r\text{ is an integer}\}
{[
a
r
]
∣
r
is an integer
}
has exactly
m
m
m
elements. Prove that there is a positive integer
q
q
q
such that
q
q
q
is coprime to
m
n
mn
mn
and
[
n
q
]
=
[
a
]
[nq]=[a]
[
n
q
]
=
[
a
]
.
number theory
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