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Problems
Contests
International Contests
International Olympiad of Metropolises
2017 IOM
5
5
Part of
2017 IOM
Problems
(1)
LCM of numbers
Source: IOM 2017 day2 p5
9/6/2017
Let
x
x
x
and
y
y
y
be positive integers such that
[
x
+
2
,
y
+
2
]
−
[
x
+
1
,
y
+
1
]
=
[
x
+
1
,
y
+
1
]
−
[
x
,
y
]
[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]
[
x
+
2
,
y
+
2
]
−
[
x
+
1
,
y
+
1
]
=
[
x
+
1
,
y
+
1
]
−
[
x
,
y
]
.Prove that one of the two numbers
x
x
x
and
y
y
y
divide the other.(Here
[
a
,
b
]
[a,b]
[
a
,
b
]
denote the least common multiple of
a
a
a
and
b
b
b
).Proposed by Dusan Djukic.
number theory