Found problems: 74
Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$
For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times.
Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.
Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.
Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.
A positive integer is called [i]decimal palindrome[/i] if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$. Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.
Determines the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.
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 $10$.
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.
Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.
Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number
\[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\]
is a multiple of $19$.
Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.
We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.
How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$.
(For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)
Find all natural numbers of $90$ digits that are multiples of $13$ and have the first $43$ digits equal to each other and nonzero, the last $43$ digits equal to each other, and the middle $4$ digits are $2, 0, 1, 0$, in that order.
Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$.
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.
1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$.
2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$.
3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?
Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$
The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$.
How many sums $$x_1+x_2+x_3, \ \ 1\leq x_j\leq 300, \ j=1,2,3$$ are multiples of $3$;
Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ?
I. Gorodnin
Given are $n$ integers. Prove that at least one of the following conditions applies:
1) One of the numbers is a multiple of $n$.
2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.