Difference between revisions of "2019 AIME I Problems/Problem 14"
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− | <math>\phi(k)</math> is the [[Euler Totient Function]] of integer <math>k</math>. <math>\phi(k)</math> is the number of positive integers less than <math>k</math> relatively prime to <math>k</math>. Define the numbers <math>k_1,k_2,k_3,\cdots,k_n</math> to be the prime factors of <math>k</math>. Then, we have < | + | <math>\phi(k)</math> is the [[Euler Totient Function]] of integer <math>k</math>. <math>\phi(k)</math> is the number of positive integers less than <math>k</math> relatively prime to <math>k</math>. Define the numbers <math>k_1,k_2,k_3,\cdots,k_n</math> to be the prime factors of <math>k</math>. Then, we have <cmath>\phi(k)=k\cdot \prod^n_{i=1}\left(1-\dfrac{1}{k_i}\right).</cmath> A property of the Totient function is that, for any prime <math>p</math>, <math>\phi(p)=p-1</math>. |
− | [[Euler's Totient Theorem]] states that <cmath>a^{\phi(k)} \equiv 1\pmod k</cmath> if < | + | [[Euler's Totient Theorem]] states that <cmath>a^{\phi(k)} \equiv 1\pmod k</cmath> if <math>\gcd(a,k)=1</math>. |
− | Furthermore, the order < | + | Furthermore, the order <math>a</math> modulo <math>n</math> for an integer <math>a</math> relatively prime to <math>n</math> is defined as the smallest positive integer <math>d</math> such that <math>a^{d} \equiv 1\pmod n</math>. An important property of the order <math>d</math> is that <math>d|\phi(n)</math>. |
==Video Solution== | ==Video Solution== |
Revision as of 19:27, 18 May 2020
Problem 14
Find the least odd prime factor of .
Solution
We know that for some prime . We want to find the smallest odd possible value of . By squaring both sides of the congruence, we find .
Since , the order of modulo is a positive divisor of .
However, if the order of modulo is or then will be equivalent to which contradicts the given requirement that .
Therefore, the order of modulo is . Because all orders modulo divide , we see that is a multiple of . As is prime, . Therefore, . The two smallest primes equivalent to are and . As and , the smallest possible is thus .
Note to solution
is the Euler Totient Function of integer . is the number of positive integers less than relatively prime to . Define the numbers to be the prime factors of . Then, we have A property of the Totient function is that, for any prime , .
Euler's Totient Theorem states that if .
Furthermore, the order modulo for an integer relatively prime to is defined as the smallest positive integer such that . An important property of the order is that .
Video Solution
On The Spot STEM:
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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