2019 AIME I Problems/Problem 14
The 2019 AIME I takes place on March 13, 2019.
Problem 14
Find the least odd prime factor of .
Solution 2 the solution in the chat is better and should be sol 1
Essentially, we are trying to find the smallest prime p such that . This congruence tells us that . Therefore, the order of 2019 modulo p is a divisor of 16, but not a divisor of 8. This tells us that the order of 2019 modulo p is exactly 16 since it is the only possibility. We know that the order of 2019 modulo p is a divisor of , which is just p-1 because p is prime.Thus, we have . Now, we just test up. 17 does not work, because reduces to 2 modulo 17. The reason this does not work is because reduces to 1, not -1 modulo 17. Since, 33, 49, 65, and 81 are all composite, we are able to skip those cases. This brings us to 97, which gives . Thus .
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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