Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.

Is there a number whose digits are only $1$'s and which is divided by $1999$?

Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera. At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins. (a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty. (b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

Do integers $a, b$ exist such that $a^{2006} + b^{2006} + 1$ is divisible by $2006^2$?

Given is a nonzero integer $k$. Prove that equation $k =\frac{x^2 - xy + 2y^2}{x + y}$ has an odd number of ordered integer pairs $(x, y)$ just when $k$ is divisible by seven.

A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure: In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.

Prove that $n^2 + 3n + 5$ is not divisible by $121$ for any positive integer $n$.

Benedek wrote the following $300 $ statements on a piece of paper. $2 | 1!$ $3 | 1! \,\,\, 3 | 2!$ $4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3!$ $5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4!$ $...$ $24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23!$ $25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24!$ How many true statements did Benedek write down? The symbol | denotes divisibility, e.g. $6 | 4!$ means that $6$ is a divisor of number $4!$.

We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

The sum of all even positive integers less than $100$ those are not divisible by $3$ is (A): $938$, (B): $940$, (C): $1634$, (D): $1638$, (E): None of the above.

Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.

(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$. (b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.

(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd. (D Fomin, St Petersburg)

What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times ... \times 50$ is divisible by $10^n$?

Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.

Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$

Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

The number $2022$ is written on the board. In each step, we replace one of the $2$ digits with the number $2022$. For example $$2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...$$ After how many steps can a number divisible by $22$ be written on the board? Specify all options.

Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.

Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.

Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$