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2017 China Northern MO
7
7
Part of
2017 China Northern MO
Problems
(1)
China Northern Mathematical Olympiad, Problem 7
Source: China Northern Mathematical Olympiad
7/29/2017
Let
S
(
n
)
S(n)
S
(
n
)
denote the sum of the digits of the base-10 representation of an natural number
n
n
n
. For example.
S
(
2017
)
=
2
+
0
+
1
+
7
=
10
S(2017) = 2+0+1+7 = 10
S
(
2017
)
=
2
+
0
+
1
+
7
=
10
. Prove that for all primes
p
p
p
, there exists infinitely many
n
n
n
which satisfy
S
(
n
)
≡
n
m
o
d
p
S(n) \equiv n \mod p
S
(
n
)
≡
n
mod
p
.
number theory
sum of digits