MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
2023 ITAMO
3
3
Part of
2023 ITAMO
Problems
(1)
NT with sum of digits
Source: Italy MO 2023 P3
5/7/2023
Let
s
(
n
)
s(n)
s
(
n
)
denote the sum of the digits of
n
n
n
. a) Do there exist distinct positive integers
a
,
b
a, b
a
,
b
, such that
2023
a
+
s
(
a
)
=
2023
b
+
s
(
b
)
2023a+s(a)=2023b+s(b)
2023
a
+
s
(
a
)
=
2023
b
+
s
(
b
)
? b) Do there exist distinct positive integers
a
,
b
a, b
a
,
b
, such that
a
+
2023
s
(
a
)
=
b
+
2023
s
(
b
)
a+2023s(a)=b+2023s(b)
a
+
2023
s
(
a
)
=
b
+
2023
s
(
b
)
?
number theory