2019 AIME I Problems/Problem 14
Problem 14
Find the least odd prime factor of .
Solution 1
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 . This tells us that 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 1
is called the "Euler Function" of integer . Eular 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 .
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2019 AIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 15 | |
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