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Saint Petersburg Mathematical Olympiad
2001 Saint Petersburg Mathematical Olympiad
9.4
9.4
Part of
2001 Saint Petersburg Mathematical Olympiad
Problems
(1)
Prove that (ab+c, bc+a, ca+b)=(a,b,c)
Source: St. Petersburg MO 2001 Grade 9 Problem 4
3/1/2023
Let
a
,
b
,
c
∈
Z
+
a,b,c\in\mathbb{Z^{+}}
a
,
b
,
c
∈
Z
+
such that
(
a
2
−
1
,
b
2
−
1
,
c
2
−
1
)
=
1
(a^2-1, b^2-1, c^2-1)=1
(
a
2
−
1
,
b
2
−
1
,
c
2
−
1
)
=
1
Prove that
(
a
b
+
c
,
b
c
+
a
,
c
a
+
b
)
=
(
a
,
b
,
c
)
(ab+c, bc+a, ca+b)=(a,b,c)
(
ab
+
c
,
b
c
+
a
,
c
a
+
b
)
=
(
a
,
b
,
c
)
(As usual,
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
means the greatest common divisor of numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
) [I]Proposed by A. Golovanov
St. Petersburg MO
GCD
number theory
Divisibility
greatest common divisor