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Philippine MO
2019 Philippine MO
3
3
Part of
2019 Philippine MO
Problems
(1)
system 3x3 in N^3 with LCM a^2 + b^2 = n lcm(a, b) + n^2 PMO 2019
Source: 21st Philippine Mathematical Olympiad 2019 p3 PMO
1/7/2020
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
a
2
+
b
2
=
n
⋅
l
c
m
(
a
,
b
)
+
n
2
a^2 + b^2 = n\cdot lcm(a, b) + n^2
a
2
+
b
2
=
n
⋅
l
c
m
(
a
,
b
)
+
n
2
b
2
+
c
2
=
n
⋅
l
c
m
(
b
,
c
)
+
n
2
b^2 + c^2 = n \cdot lcm(b, c) + n^2
b
2
+
c
2
=
n
⋅
l
c
m
(
b
,
c
)
+
n
2
c
2
+
a
2
=
n
⋅
l
c
m
(
c
,
a
)
+
n
2
c^2 + a^2 = n \cdot lcm(c, a) + n^2
c
2
+
a
2
=
n
⋅
l
c
m
(
c
,
a
)
+
n
2
for some positive integer
n
n
n
.
system of equations
number theory
least common multiple