Found problems: 408
Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.
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 $16$?
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 $m$.
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$.
What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ?
(S Tokarev)
We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?
Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$
Let be given a set $\{r_1,r_2,...,r_k\}$ of natural numbers that give distinct remainders when divided by a natural number $m$. Prove that if $k > m/2$, then for every integer $n$ there exist indices $i$ and $j$ (not necessarily distinct) such that $r_i +r_j -n$ is divisible by $m$.
Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property:
[i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i]
Prove that this polynomial is divisible by $x$.
Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?
Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$.
(a) Show that $s(n)$ is an integer whenever $n$ is an integer.
(b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?
How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
How many positive integers $x$ less than $10 000$ are there such that $2^x - x^2$ is divisible by $7$ ?
Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.
Prove that for any integer $n$ the number $n (n^2 + 5)$ is divisible by $6$.
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 $2017$.
Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.
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 $m$.
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)
The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ?
Mikhail Evdokimov
Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$?
(AI Galochkin)
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?