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2014 Harvard-MIT Mathematics Tournament
21
21
Part of
2014 Harvard-MIT Mathematics Tournament
Problems
(1)
2014 Guts #21: Sum of Powers of Two Divisible by Five
Source:
4/21/2014
Compute the number of ordered quintuples of nonnegative integers
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
(a_1,a_2,a_3,a_4,a_5)
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
such that
0
≤
a
1
,
a
2
,
a
3
,
a
4
,
a
5
≤
7
0\leq a_1,a_2,a_3,a_4,a_5\leq 7
0
≤
a
1
,
a
2
,
a
3
,
a
4
,
a
5
≤
7
and
5
5
5
divides
2
a
1
+
2
a
2
+
2
a
3
+
2
a
4
+
2
a
5
2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}
2
a
1
+
2
a
2
+
2
a
3
+
2
a
4
+
2
a
5
.