2019 AIME I Problems/Problem 14
Find the least odd prime factor of .
The problem tells us that for some prime . We want to find the smallest odd possible value of . By squaring both sides of the congruence, we get .
Since , = or
However, if = or then clearly will be instead of , causing a contradiction.
Therefore, . Because , is a multiple of 16. Since we know is prime, or . Therefore, must be . The two smallest primes that are are and . , but , so our answer is .
Note to solution
is called the Euler Totient Function of integer . Euler's Totient Theorem: define as the number of positive integers less than but relatively prime to , then we have where are the prime factors of . Then, we have if .
Furthermore, for an integer relatively prime to is defined as the smallest positive integer such that . An important property of the order is that .
On The Spot STEM:
1. Take remainder of prime
2. Take that to the power of , mod
3. If it is congruent to 1, you are done
4. If not, repeat for next prime
5. Through 2 hours and 59 minutes of bashing, we arrive at the answer of -Trex4days
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