Found problems: 408
Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.
a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$.
b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?
Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$.
(A Galochkin)
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.
Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$
then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.
Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.
Put $P = m^{2003}n^{2017} - m^{2017}n^{2003}$ , where $m, n \in N$.
a) Is $P$ divisible by $24$?
b) Do there exist $m, n \in N$ such that $P$ is not divisible by $7$?
The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.
Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.
Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.
Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.
Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements.
Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.
The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows:
$$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$
Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.
A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.
Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?
The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$.
Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$.
[i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".
Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$
Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.
We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.
Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.
Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other.
(An established theorem)
All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?