Found problems: 471
Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5$ if and only if $n$ is not divisible by $4$.
Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that
\[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]
The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
Find all prime numbers $p$ such that
$(x + y)^{19} - x^{19} - y^{19}$
is a multiple of $p$ for any positive integers $x$, $y$.
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
Prove that for $n>1$ , $n$ does not divide $2^{n-1}+1$
Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.
Let $a$ be an integer and $n$ a positive integer . Show that the sum :
$$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .
Let $a$ and $n$ be positive integers such that:
1. $a^{2^n}-a$ is divisible by $n$,
2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is [i]not[/i] divisible by $n$.
Prove that $n$ has a prime factor [i]smaller[/i] than $2024$.
[i]Proposed by Shantanu Nene[/i]
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\].
[i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$.
Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$.
[i]AMM Magazine[/i]
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
What is the units digit of $ 13^{2003}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
Find all nonnegative integers $x,y$ such that \[ 2 \cdot 3^{x} +1 = 7 \cdot 5^{y}. \]
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$
a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$
b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$
Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.
Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers.
$(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number.
$(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.
For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$
(a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$
(b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$
[I]United Kingdom[/i]