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PEN D Problems
19
D 19
D 19
Source:
May 25, 2007
modular arithmetic
Congruences
Problem Statement
Let
a
1
a_{1}
a
1
,
⋯
\cdots
⋯
,
a
k
a_{k}
a
k
and
m
1
m_{1}
m
1
,
⋯
\cdots
⋯
,
m
k
m_{k}
m
k
be integers with
2
≤
m
1
2 \le m_{1}
2
≤
m
1
and
2
m
i
≤
m
i
+
1
2m_{i}\le m_{i+1}
2
m
i
≤
m
i
+
1
for
1
≤
i
≤
k
−
1
1 \le i \le k-1
1
≤
i
≤
k
−
1
. Show that there are infinitely many integers
x
x
x
which do not satisfy any of congruences
x
≡
a
1
(
m
o
d
m
1
)
,
x
≡
a
2
(
m
o
d
m
2
)
,
⋯
,
x
≡
a
k
(
m
o
d
m
k
)
.
x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.
x
≡
a
1
(
mod
m
1
)
,
x
≡
a
2
(
mod
m
2
)
,
⋯
,
x
≡
a
k
(
mod
m
k
)
.
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