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Problems
Contests
National and Regional Contests
Lithuania Contests
Lithuania National Olympiad
2010 Lithuania National Olympiad
4
divisible by 37
divisible by 37
Source: Lithuania NMO 2010
March 11, 2012
modular arithmetic
number theory proposed
number theory
Problem Statement
Decimal digits
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
37
∣
(
a
0
a
0
…
a
0
b
0
c
0
c
…
0
c
)
10
37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10}
37
∣
(
a
0
a
0
…
a
0
b
0
c
0
c
…
0
c
)
10
where there are
1001
1001
1001
a's and
1001
1001
1001
c's. Prove that
b
=
a
+
c
b=a+c
b
=
a
+
c
.
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