2014 AIME II Problems/Problem 15
Problem
For any integer , let
be the smallest prime which does not divide
Define the integer function
to be the product of all primes less than
if
, and
if
Let
be the sequence defined by
, and
for
Find the smallest positive integer
such that
Solution
Note that for any , for any prime
,
. This provides motivation to translate
into a binary sequence
.
Let the prime factorization of be written as
, where
is the
th prime number. Then, for every
in the prime factorization of
, place a
in the
th digit of
. This will result in the conversion
.
Multiplication for the sequence will translate to addition for the sequence
. Thus, we see that
translates into
. Since
, and
,
corresponds to
, which is
in binary. Since
,
=
.
Solution 2 (Painful Bash)
We go through the terms and look for a pattern. We find that
Commit to the bash. Eventually, you will receive that , so
is the answer. Trust me, this is worth the 10 index points.
See also
2014 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
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