Found problems: 387
Determine the largest integer $k$ for which $12^k$ is a factor of $120! $
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.
Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.
Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .
A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.
Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that
$m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.
Given $26$ different positive integers , in any six numbers of the $26$ integers , there are at least two numbers , one can be devided by another. Then prove : There exists six numbers , one of them can be devided by the other five numbers .
Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.
Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers?
(Folklore)
Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.
Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $.
Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .
Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.
Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers.
( M . N . Gusarov, Leningrad)
Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.
Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.
Find positive integers n that satisfy the following two conditions:
(a) the quotient obtained when $n$ is divided by $9$ is a positive three digit number, that has equal digits.
(b) the quotient obtained when $n + 36$ is divided by $4$ is a four digit number, the digits beeing $2, 0, 0, 9$ in some order.
A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.
Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.
Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.
Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions:
$x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$
Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.