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International Contests
IberoAmerican
2005 IberoAmerican
4
4
Part of
2005 IberoAmerican
Problems
(1)
Sum of remainders
Source: Iberoamerican Olympiad 2005
9/29/2005
Denote by
a
m
o
d
b
a \bmod b
a
mod
b
the remainder of the euclidean division of
a
a
a
by
b
b
b
. Determine all pairs of positive integers
(
a
,
p
)
(a,p)
(
a
,
p
)
such that
p
p
p
is prime and
a
m
o
d
p
+
a
m
o
d
2
p
+
a
m
o
d
3
p
+
a
m
o
d
4
p
=
a
+
p
.
a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p.
a
mod
p
+
a
mod
2
p
+
a
mod
3
p
+
a
mod
4
p
=
a
+
p
.
number theory solved
number theory