Given $100$ integers, is it always possible to choose $15$ of them such that the difference of any two of the chosen numbers is divisible by $7$? What is the answer if $15$ is replaced by $16$?

Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

The six-digit decimal number contains six different non-zero digits and is divisible by $37$. Prove that having transposed its digits you can obtain at least $23$ more numbers divisible by $37$

Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$).

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$

Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.

Prove that there exists an innite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

Find six distinct positive integers such that the product of any two of them is divisible by their sum. (D. Fomin, Leningrad)

Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$

How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?

For positive integers $x,y$, find all pairs $(x,y)$ such that $x^2y + x$ is a multiple of $xy^2 + 7$.

We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that: $\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$. $\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$. Find the maximum value of $n$

Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$