Found problems: 408
Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?
It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers
Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers.
Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.
Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that:
(i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and
(ii) each $b_i$ is a product of one or more terms of $(a_i)$.
How many numbers $\overline{abcd}$ with different digits satisfy the following property:
if we replace the largest digit with the digit $1$ results in a multiple of $30$?
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.
Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?
Let $p$ be an odd prime number.
a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n.
b) Find all $n$ satisfy condition above when $p = 3$
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
The number $2022$ is written on the board. In each step, we replace one of the $2$ digits with the number $2022$.
For example $$2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...$$
After how many steps can a number divisible by $22$ be written on the board? Specify all options.
Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.
Determine all pairs $(a, b)$ of integers having the following property:
there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.
Let $a,b, c,d$ be four integers. Prove that the product of the six differences
$$b - a,c - a,d - a,d - c,d - b, c - b$$
is divisible by $12$.
Let $n$ be a positive integer. Show that
$${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.
Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?
Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that
$$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.
Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties:
1) $1 \le ai \le p$ for all $i = 1, . . . , p$.
2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$.
3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$.
[i](Walther Janous)[/i]
From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.
Let $a, b$ and $c$ be integers such that $a + b + c = 0$.
(a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$.
(b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$.
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Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.