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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