Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

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 $n$.

Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?

Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.

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

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

Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

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 $n + 3$.

Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.

Let $a$ and $b$ be distinct positive integers, bigger that $10^6$, such that $(a+b)^3$ is divisible with $ab$. Prove that $ \mid a-b \mid > 10^4$

Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.

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

Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

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

Prove that among any $k + 1$ natural numbers there are two numbers whose difference is divisible by $k$.

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]

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

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 $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.