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Princeton University Math Competition
2018 Princeton University Math Competition
2018 PUMaC Combinatorics B
2018 PUMaC Combinatorics B
Part of
2018 Princeton University Math Competition
Subcontests
(2)
4
1
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2018 PUMaC Combinatorics B4
Let
N
N
N
be the number of sequences of natural numbers
d
1
,
d
2
,
…
,
d
10
d_1,d_2,\dots,d_{10}
d
1
,
d
2
,
…
,
d
10
such that the following conditions hold:
d
1
∣
d
2
d_1|d_2
d
1
∣
d
2
,
…
\dots
…
,
d
9
∣
d
10
d_9|d_{10}
d
9
∣
d
10
and
d
10
∣
6
2018
d_{10}|6^{2018}
d
10
∣
6
2018
. Evaluate the remainder when
N
N
N
is divided by
2017
2017
2017
.
1
1
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2018 PUMaC Combinatorics B1
You have four fair
6
6
6
-sided dice, each numbered
1
1
1
to
6
6
6
(inclusive). If all four dice are rolled, the probability that the product of the rolled numbers is prime can be written as
a
b
\tfrac{a}{b}
b
a
, where
a
a
a
and
b
b
b
are relatively prime. What is
a
+
b
a+b
a
+
b
?