Found problems: 408
A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$?
(M . I. Gusarov)
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.
Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.
Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.
Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.
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 $169$.
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 $a$ .
Show that the number $5^{2n+1}2^{n+2}+3^{n+2}2^{2n+1}$ is divisible by $19$ for every non-negative integer $n$.
Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.
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 $(n^n)!$ .
A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?
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)
Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$.
Remember, integers may be positive, negative, or zero!
(a) Prove that $N$ is odd.
(b) Prove that $N$ is not divisible by $3$.
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.