Found problems: 408
Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.
Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.
How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.
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
Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.
Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.
Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.
For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?
Determine all positive integer numbers $n$ satisfying the following condition:
the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.
Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?
Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.
$p$ is a prime and $m$ is a non-negative integer $< p-1$.
Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.
For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order?
(A Shapovalov)
Prove that the number $9^n+8^n+7^n+6^n-4^n-3^n-2^n-1^n$ is divisible by $10$ for all non-negative $n$.
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 $7$?
Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.
Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$
Is there an integer $n$ such that $n^2 + n + 1$ is divisible by $1955$ ?