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Baltic Way
2016 Baltic Way
4
Divisibility
Divisibility
Source: Baltic Way 2016, Problem 4
November 5, 2016
number theory
Problem Statement
Let
n
n
n
be a positive integer and let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be integers such that
n
∣
a
+
b
+
c
+
d
n | a + b + c + d
n
∣
a
+
b
+
c
+
d
and
n
∣
a
2
+
b
2
+
c
2
+
d
2
.
n | a^2 + b^2 + c^2 + d^2.
n
∣
a
2
+
b
2
+
c
2
+
d
2
.
Show that
n
∣
a
4
+
b
4
+
c
4
+
d
4
+
4
a
b
c
d
.
n | a^4 + b^4 + c^4 + d^4 + 4abcd.
n
∣
a
4
+
b
4
+
c
4
+
d
4
+
4
ab
c
d
.
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