2021 AIME I Problems/Problem 14

Problem

For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$ . Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$ . Find the sum of the prime factors in the prime factorization of $n$ .

Solution

We first claim that $n$ must be divisible by 42. Since $\sigma(a^n)-1$ is divisible by 2021 for all positive integers $a$ , we can first consider the special case where $a \neq 0,1 \pmod{43}$ .

Then $\sigma(a^n)-1 = \sum_{i=1}^n a^i = a\left(\frac{a^n - 1}{a-1}\right)$ . In order for this expression to be divisible by $2021=43\times 47$ , a necessary condition is $a^n - 1 \equiv 0 \pmod{43}$ . By Fermat's Little Theorem, $a^{42} \equiv 1 \pmod{43}$ . Moreover, if $a$ is a primitive root modulo 43, then $\text{ord}_{43}(a) = 42$ , so $n$ must be divisible by 42.

By similar reasoning, $n$ must be divisible by 46, by considering $a \neq 0,1 \pmod{47}$ .

We next claim that $n$ must be divisible by 43 and 47. Consider the case $a=2022$ . Then $\sigma(a^n) \equiv n \pmod{2021}$ , so $\sigma(2022^n)-1$ is divisible by 2021 if and only if $n$ is divisible by 2021.

Lastly, we claim that if $n = \text{lcm}(42, 46, 43, 47)$ , then $\sigma(a^n) - 1$ is divisible by 2021 for all positive integers $a$ . The claim is trivially true for $a=1$ so suppose $a>1$ . Let $a = p_1^{e_1}\ldots p_k^{e_k}$ be the prime factorization of $a$ . Since $\sigma$ is multiplicative, we have $\sigma(a^n) - 1 = \prod_{i=1}^k \sigma(p_i^{e_in}) - 1.$ We can show that $\sigma(p_i^{e_in}) \equiv 1 \pmod{2021}$ for all primes $p_i$ and integers $e_i \ge 1$ , where $\sigma(p_i^{e_in}) = 1 + (p_i + p_i^2 + \ldots + p_i^n) + (p_i^{n+1} + \ldots + p_i^{2n}) + \ldots + (p_i^{n(e_i-1)+1} + \ldots + p_i^{e_in})$ where each expression in parentheses contains $n$ terms. It is easy to verify that if $p_i = 43$ or $p_i = 47$ then $\sigma(p_i^{e_in}) \equiv 1 \pmod{2021}$ for this choice of $n$ , so suppose $p_i \not\equiv 0 \pmod{43}$ and $p_i \not\equiv 0 \pmod{47}$ . Each expression in parentheses equals $\frac{p_i^n - 1}{p_i - 1}$ multiplied by some power of $p_i$ . If $p_i \neq 1 \pmod{43}$ , then FLT implies $p_i^n - 1 \equiv 0 \pmod{43}$ , and if $p_i \equiv 1 \pmod{43}$ , then $p_i + p_i^2 + \ldots + p_i^n \equiv 1 + 1 + \ldots + 1 \equiv 0 \pmod{43}$ (since $n$ is also a multiple of 43, by definition). Similarly, we can show $\sigma(p_i^{e_in}) \equiv 1 \pmod{47}$ , and a simple CRT argument shows $\sigma(p_i^{e_in}) \equiv 1 \pmod{2021}$ . Then $\sigma(a^n) \equiv 1^k \equiv 1 \pmod{2021}$ .

Then the prime factors of $n$ are $2$ , $3$ , $7$ , $23$ , $43$ , and $47$ , and the answer is $2+3+7+23+43+47 = \boxed{125}$ . ~scrabbler94

Solution 2 (cheap and not very reliable)

Since the problem works for all positive integers $a$ , let's plug in $a=2$ and see what we get. Since $\sigma({2^n}) = 2^{n+1}-1,$ we have $2^{n+1} \equiv 2 \pmod{2021}.$ Simplifying using CRT and Fermat's Little Theorem, we get that $2^n \equiv 0 \pmod{42}$ and $2^n \equiv 0 \pmod{46}.$ Then, we can look at $a=2022$ just like in Solution 1 to find that $43$ and $47$ also divide $n$ . There don't seem to be any other odd "numbers" to check, so we can hopefully assume that the answer is the sum of the prime factors of $\text{lcm(42, 43, 46, 47)}.$ From here, follow solution 1 to get the final answer.

-PureSwag

Solution 3 (Kind of similar to Sol 2 and kind of what cosmicgenius said)

It says the problem implies that it works for all positive integers $a$ , we basically know that If $p \equiv 1 \pmod q$ , than from "USEMO 2019 Problem 4 which was $p^n + p^{n-1} + ... + 1 \equiv 1$ (mod $q$ ) we have that $\frac{p^{en+1}-1}{p-1} \equiv p^{en} + p^{en-1} + \dots + 1 \equiv en + 1 \equiv 1 \pmod q,$ . From here we can just let $\sigma(2^n)$ or be a power of 2 which we can do $\sigma(2^n)=1+2+2^2+2^3+2^4+\cdots+2^n=2^{n+1}-1$ which is a geo series. We can plug in $a=2^a$ like in Solution 2 and use CRT which when we prime factorize we get that $2021 = 43 \cdot 47$ which like everyone knows. We use CRT to find that \begin{align*} 2^n &\equiv 1 \pmod{43}, \\ 2^n &\equiv 1 \pmod{47}. \end{align*}. We see that this is just FLT which is $a^{p-1} \equiv 1 \pmod p$ we see all multiples of 42 will work for first and 46 for the second. We can figure out that it is just $\text{lcm}(43-1,47-1)\cdot43\cdot47$ which when we add up we get that it's just the sum of the prime factors of lcm(42,43,46,47) which you can just look at Solution 1 to find out the sum of the prime factors and get the answer.

Remarks: This was kind of used in what cosmicgenius and Solution 2 said and what mathisawesome2169 said and It "Reminded me of 2019 USEMO Problem 4 when solving the $p^n + p^{n-1} + ... + 1 \equiv 1$ which gave $p \equiv 1$ $q$ . Sorry if it's bad I need to do something else so I kind of rushed.

Solution 4 (Along the line of Solution 1)

$n$ only needs to satisfy $\sigma(a^n)\equiv 1 \pmod{43}$ and $\sigma(a^n)\equiv 1 \pmod{47}$ for all $a$ . Let's work on the first requirement (mod 43) first. All $n$ works for $a=1$ . If $a>1$ , let $a$ 's prime factorization be $a=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ . The following 3 statements are the same. $\text{for all }a, \sigma(a^n)=\prod_{i=1}^k(1+p_i+\cdots+p_i^{ne_i})\equiv1\pmod{43}$ $\text{for all }e>0\text{ and prime }p, 1+p+\cdots+p^{ne}\equiv1\pmod{43}$ $\text{for all prime }p, p+p^2+\cdots+p^n \equiv 0 \pmod{43}$

(1) For $p\equiv0\pmod{43}$ , no matter $n$ , it is true $p+p^2+\cdots+p^n \equiv 0 \pmod{43}$

(2) For all $p\equiv1\pmod{43}$ , $p+p^2+\cdots+p^n \equiv n \equiv 0 \pmod{43}$ One can either use brute force or Dirichlet's Theorem to show such $p$ exists. Therefore $43|n.$

(3) For all $p\not\equiv0,1\pmod{43}$ , $p+p^2+\cdots+p^n \equiv 0 \pmod{43} \Leftrightarrow p^n-1\equiv0\pmod{43}$ According to Fermat's Little Theorem, $42|n$ is sufficient. To show it's necessary we just need to show 43 has a prime primitive root. This can be done either by brute force or as follows. 43 is prime and it must have a primitive root $t\neq 1$ that's coprime to 43. All numbers of the form $43k+t$ are also primitive roots of 43. According to Dirichlet's Theorem there must be primes among them.

Similar arguments for (mod 47) lead to $46|n$ and $47|n$ . Therefore $n=\text{lcm}[42,43,46,47]$

2021 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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