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International Contests
Austrian-Polish
1991 Austrian-Polish Competition
8
8
Part of
1991 Austrian-Polish Competition
Problems
(1)
xy=- 1 (mod z), yz = 1 (mod x), zx = 1 (mod y).
Source: Austrian Polish 1991 APMC
5/1/2020
Consider the system of congruences
{
x
y
≡
−
1
(
m
o
d
z
)
y
z
≡
1
(
m
o
d
x
)
z
x
≡
1
(
m
o
d
y
)
\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}
⎩
⎨
⎧
x
y
≡
−
1
(
m
o
d
z
)
yz
≡
1
(
m
o
d
x
)
z
x
≡
1
(
m
o
d
y
)
Find the number of triples
(
x
,
y
,
z
)
(x,y, z)
(
x
,
y
,
z
)
of distinct positive integers satisfying this system such that one of the numbers
x
,
y
,
z
x,y, z
x
,
y
,
z
equals
19
19
19
.
number theory
system of equations