Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor? $ \textbf{(A)}\ p^8q^8r^8\qquad \textbf{(B)}\ (pq^2r^2)^3\qquad \textbf{(C)}\ (p^2q^2r^2)^3\qquad \textbf{(D)}\ (pqr^2)^3\qquad \textbf{(E)}\ 4p^3q^3r^3$

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

Let $p$ be a prime number. At a school of $p^{2020}$ students it is required that each club consist of exactly $p$ students. Is it possible for each pair of students to have exactly one club in common?

Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.

Let $p$ be some prime number. a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$. b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.

Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well. [i]Proposed by Valentio Iverson (Indonesia)[/i]

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

Prove that there are infinitely many positive $n$ that for all prime divisors $p$ of $n^2 + 3, \exists 0 \leq k \leq \sqrt{n}$ and $p \mid k^2+3$

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

Professor Moriarty has designed a “prime-testing trail.” The trail has $2002$ stations, labeled $1,... , 2002$. Each station is colored either red or green, and contains a table which indicates, for each of the digits $0, ..., 9$, another station number. A student is given a positive integer $n$, and then walks along the trail, starting at station $1$. The student reads the first (leftmost) digit of $n,$ and looks this digit up in station $1$’s table to get a new station location. The student then walks to this new station, reads the second digit of $n$ and looks it up in this station’s table to get yet another station location, and so on, until the last (rightmost) digit of $n$ has been read and looked up, sending the student to his or her final station. Here is an example that shows possible values for some of the tables. Suppose that $n = 19$: [img]https://cdn.artofproblemsolving.com/attachments/f/3/db47f6761ca1f350e39d53407a1250c92c4b05.png[/img] Using these tables, station $1$, digit $1$ leads to station $29$m station $29$, digit $9$ leads to station $1429$, and station $1429$ is green. Professor Moriarty claims that for any positive integer $n$, the final station (in the example, $1429$) will be green if and only if $n$ is prime. Is this possible?

Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$