Found problems: 408
Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$.
(BR Frenkin)
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.
Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$.
Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.
Prove that $n^2 + 3n + 5$ is not divisible by $121$ for any positive integer $n$.
For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$.
Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.
Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.
The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.
Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.
The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.
Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above.
How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?
Find all pairs $ (m, n)$ of positive integers that have the following property:
For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$
Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd.
(D Fomin, St Petersburg)
Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.
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 $n + 3$.
Find all pairs $(x, y)$ of integers satisfying the following condition:
each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ .
(S Zlobin)
The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?
Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.
Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$).
Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
Show that in any set of three distinct integers there are two of them, say $a$ and $b$ such that the number $a^5b^3-a^3b^5$ is a multiple of $10$.
For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.
Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.