2019 AIME I Problems/Problem 14

Problem 14

Find the least odd prime factor of $2019^8+1$ .

Solution

We know that $2019^8 \equiv -1 \pmod{p}$ for some prime $p$ . We want to find the smallest odd possible value of $p$ . By squaring both sides of the congruence, we find $2019^{16} \equiv 1 \pmod{p}$ .

Since $2019^{16} \equiv 1 \pmod{p}$ , the order of $2019$ modulo $p$ is a positive divisor of $16$ .

However, if the order of $2019$ modulo $p$ is $1, 2, 4,$ or $8,$ then $2019^8$ will be equivalent to $1 \pmod{p},$ which contradicts the given requirement that $2019^8\equiv -1\pmod{p}$ .

Therefore, the order of $2019$ modulo $p$ is $16$ . Because all orders modulo $p$ divide $\phi(p)$ , we see that $\phi(p)$ is a multiple of $16$ . As $p$ is prime, $\phi(p) = p\left(1 - \dfrac{1}{p}\right) = p - 1$ . Therefore, $p\equiv 1 \pmod{16}$ . The two smallest primes equivalent to $1 \pmod{16}$ are $17$ and $97$ . As $2019^8 \not\equiv -1 \pmod{17}$ and $2019^8 \equiv -1 \pmod{97}$ , the smallest possible $p$ is thus $\boxed{097}$ .

Note to solution

$\phi(k)$ is the Euler Totient Function of integer $k$ . $\phi(k)$ is the number of positive integers less than $k$ relatively prime to $k$ . Define the numbers $k_1,k_2,k_3,\cdots,k_n$ to be the prime factors of $k$ . Then, we have $\phi(k)=k\cdot \prod^n_{i=1}\left(1-\dfrac{1}{k_i}\right).$ $A property of the Totient function is that, for any prime$ p $,$ \phi(p)=p-1$.

[[Euler's Totient Theorem]] states that <cmath>a^{\phi(k)} \equiv 1\pmod k</cmath> if$ (Error compiling LaTeX. Unknown error_msg)\gcd(a,k)=1$.

Furthermore, the order$ (Error compiling LaTeX. Unknown error_msg)a $modulo$ n $for an integer$ a $relatively prime to$ n $is defined as the smallest positive integer$ d $such that$ a^{d} \equiv 1\pmod n $. An important property of the order$ d $is that$ d|\phi(n)$.

Video Solution

On The Spot STEM:

https://youtu.be/_vHq5_5qCd8

https://youtu.be/IF88iO5keFo

2019 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2019 AIME I Problems/Problem 14

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Problem 14

Solution

Note to solution

Video Solution

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