This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 367

2015 Singapore Junior Math Olympiad, 1
Tags: multiple , number theory , divides , Digits
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 $37$.

1998 Singapore Senior Math Olympiad, 1
Tags: Integer , divisible , divides , number theory
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

1995 Singapore MO Open, 4
Tags: number theory , divisible , divides
Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

1993 Czech And Slovak Olympiad IIIA, 1
Tags: divides , divisible , number theory
Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$

2018 Saudi Arabia BMO TST, 3
Tags: divides , divisor , number theory
Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

2000 Tournament Of Towns, 3
Tags: least common multiple , LCM , number theory , Sum , Product , divisible , divides
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

2019 Saudi Arabia Pre-TST + Training Tests, 5.1
Tags: number theory , prime , divides , divisible
Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

1996 Singapore Team Selection Test, 3
Tags: Sequence , number theory , divides , divisible
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 $i = 1, 2, ... , 1994$

2016 Czech-Polish-Slovak Junior Match, 3
Tags: combinatorial geometry , combinatorics , divides , geometry , 3D geometry , prism
Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

2017 Switzerland - Final Round, 4
Tags: number theory , divides , divisible
Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$ $$n + 2 | n^p + q^p.$$ Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.

1993 All-Russian Olympiad Regional Round, 9.2
Tags: number theory , divides , divisible
Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

2020 Kazakhstan National Olympiad, 1
Tags: number theory , divides
Find all pairs $ (m, n) $ of natural numbers such that $ n ^ 4 \ | \ 2m ^ 5 - 1 $ and $ m ^ 4 \ | \ 2n ^ 5 + 1 $.

1985 Czech And Slovak Olympiad IIIA, 6
Tags: number theory , divides
Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.

2014 Junior Balkan Team Selection Tests - Moldova, 4
Tags: number theory , divides , divisible
A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.

VMEO III 2006 Shortlist, N4
Tags: divisible , Digits , number theory , divides
Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.

2002 Portugal MO, 4
Tags: number theory , divides , Digits
The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

2004 Estonia National Olympiad, 4
Tags: divides , divisible , odd , number theory
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.

2006 Peru MO (ONEM), 3
Tags: number theory , divides
A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$

2023 Assara - South Russian Girl's MO, 2
Tags: number theory , divisible , divides
The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

2016 Saudi Arabia Pre-TST, 1.4
Tags: number theory , coprime , divides , divisible
Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$. 1. Show that $F(r)$ is a positive integer for any prime $r \ne p$. 2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$. 3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.

1994 Austrian-Polish Competition, 7
Tags: number theory , divides , divisible
Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

2014 Saudi Arabia GMO TST, 2
Tags: Sum of powers , number theory , divides , divisible
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$

2012 Thailand Mathematical Olympiad, 7
Tags: number theory , greatest common divisor , GCD , divides
Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

1990 Greece National Olympiad, 3
Tags: divides , divisible , number theory
For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

2019 Paraguay Mathematical Olympiad, 4
Tags: number theory , divisible , divides
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.