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.

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Found problems: 74

2019 New Zealand MO, 4
Tags: divide , divisible , number theory , multiple
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

1999 Chile National Olympiad, 5
Tags: multiple , number theory
Consider the numbers $x_1, x_2,...,x_n$ that satisfy: $\bullet$ $x_i \in \{-1,1\}$, with $i = 1, 2,...,n$ $\bullet$ $x_1x_2x_3x_4 + x_2x_3x_4x_5 +...+ x_nx_1x_2x_3 = 0$ Prove that $n$ is a multiple of $4$.

2017 Saudi Arabia BMO TST, 3
Tags: digit , multiple , number theory
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 $30$?

2012 NZMOC Camp Selection Problems, 4
Tags: divide , multiple , divisible , number theory , prime
A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2020 HK IMO Preliminary Selection Contest, 8
Tags: multiple , number theory
Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$.

2022 APMO, 1
Tags: multiple , divisible , number theory
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.

1910 Eotvos Mathematical Competition, 2
Tags: divide , multiple , number theory
Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2018 Singapore Junior Math Olympiad, 1
Tags: divide , digit , multiple , number theory
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 $37$.

1993 Spain Mathematical Olympiad, 4
Tags: prime , multiple , number theory
Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.

2011 Peru MO (ONEM), 1
Tags: consecutive , multiple , number theory
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 $n$.

1995 Bundeswettbewerb Mathematik, 4
Tags: digit , multiple , number theory
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.

2024 Polish Junior MO Finals, 5
Tags: multiple , divisibility , digit , number theory
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 $19$.

1990 IMO Shortlist, 20
Tags: pigeonhole principle , divisibility , digit , decimal representation , multiple , number theory
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

2015 Singapore Junior Math Olympiad, 1
Tags: multiple , divide , digit , number theory
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 $37$.

2013 Danube Mathematical Competition, 2
Tags: multiple , sum , combinatorics , number theory
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 $2013$

2007 QEDMO 4th, 11
Tags: set , multiple , number theory
Let $S_{1},$ $S_{2},$ $...,$ $S_{n}$ be finitely many subsets of $\mathbb{N}$ such that $S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}.$ Prove that there exists some $k\in\left\{ 1,2,...,n\right\} $ such that for each positive integer $m,$ the set $S_{k}$ contains infinitely many multiples of $m.$

2019 Durer Math Competition Finals, 6
Tags: number theory , multiple , divide , divisible , digit
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

1988 Spain Mathematical Olympiad, 3
Tags: multiple , number theory
Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$) is a multiple of $41$, then so is the other.

2003 May Olympiad, 3
Tags: multiple , sum of digits , number theory
Find the smallest positive integer that ends in $56$, is a multiple of $56$, and has the sum of its digits equal to $56$.

2009 Grand Duchy of Lithuania, 1
Tags: multiple , number theory , sum of digits
The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$. (a) Find one such number. (b) Find the smallest such number.

2004 Korea Junior Math Olympiad, 3
Tags: multiple , prime number , number theory
For an arbitrary prime number $p$, show that there exists infinitely many multiples of $p$ that can be expressed as the form $$\frac{x^2+y+1}{x+y^2+1}$$ Where $x, y$ are some positive integers.

2018 Auckland Mathematical Olympiad, 1
Tags: digit , multiple , number theory
Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.

2009 Junior Balkan Team Selection Tests - Romania, 1
Tags: multiple , number theory , digit
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 $\{a_n | n \in N*\}$.

2007 Portugal MO, 3
Tags: multiple , number theory
Determines the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.

2011 Belarus Team Selection Test, 1
Tags: divisible , multiple , number theory , combinatorics
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