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Korea National Olympiad
2016 Korea National Olympiad
7
2016 Korea MO P7 - Simple NT
2016 Korea MO P7 - Simple NT
Source: 2016 KMO Senior #7
November 12, 2016
number theory
Problem Statement
Let
N
=
2
a
p
1
b
1
p
2
b
2
…
p
k
b
k
N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}
N
=
2
a
p
1
b
1
p
2
b
2
…
p
k
b
k
. Prove that there are
(
b
1
+
1
)
(
b
2
+
1
)
…
(
b
k
+
1
)
(b_1+1)(b_2+1)\ldots(b_k+1)
(
b
1
+
1
)
(
b
2
+
1
)
…
(
b
k
+
1
)
number of
n
n
n
s which satisfies these two conditions.
n
(
n
+
1
)
2
≤
N
\frac{n(n+1)}{2}\le N
2
n
(
n
+
1
)
≤
N
,
N
−
n
(
n
+
1
)
2
N-\frac{n(n+1)}{2}
N
−
2
n
(
n
+
1
)
is divided by
n
n
n
.
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