Prove that for every nonnegative integer $ k$ there is a product $$(k + 1)(k + 2)...(k + 1980)$$ divisible by $ 1980^{197}$.

Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]

We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ? I. Gorodnin

Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.

Determine all pairs of positive integers $(m,n)$ such that m is but divisible by every integer from $1$ to $n$ (inclusive), but not divisible by $n + 1, n + 2$, and $n + 3$.

a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$. b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$

Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the first $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?

On blackboard there are $10$ different positive integers which sum is equal to $62$. Prove that product of those numbers is divisible with $60$

Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.

Let $n$ and $k$ be positive integers for which we have $4$ statements: $i)$ $n+1$ is divisible with $k$ $ii)$ $n=2k+5$ $iii)$ $n+k$ is divisible with $3$ $iv)$ $n+7k$ is prime Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.

Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.