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

2000 Tuymaada Olympiad, 1
Tags: number theory , digit , multiple , maximum
Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?

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

2019 Tournament Of Towns, 4
Tags: number theory , consecutive , multiple , sum of squares , circles
There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

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

1910 Eotvos Mathematical Competition, 2
Tags: number theory , multiple , divide
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 $u$.

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

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

2022 New Zealand MO, 5
Tags: number theory , multiple , divide , divisible , recurrence relation
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

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

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

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

2011 QEDMO 9th, 4
Tags: factorial , number theory , multiple , divide
Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

2020 China Northern MO, P5
Tags: number theory , digit , multiple
Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

2004 Korea Junior Math Olympiad, 3
Tags: number theory , prime number , multiple
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.

2020 HK IMO Preliminary Selection Contest, 15
Tags: number theory , multiple
How many ten-digit positive integers consist of ten different digits and are divisible by $99$?

2000 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , odd , multiple , last digit
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.

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

1994 Greece National Olympiad, 4
Tags: number theory , multiple
How many sums $$x_1+x_2+x_3, \ \ 1\leq x_j\leq 300, \ j=1,2,3$$ are multiples of $3$;

2013 IMAC Arhimede, 1
Tags: number theory , multiple , divisible
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 $10$.

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

2004 Junior Balkan Team Selection Tests - Moldova, 5
Tags: number theory , multiple , power of 5
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 $167$.

1927 Eotvos Mathematical Competition, 1
Tags: number theory , multiple , divide
Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$ Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.

2014 Singapore Junior Math Olympiad, 1
Tags: number theory , multiple , digit
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

1999 Chile National Olympiad, 5
Tags: number theory , multiple
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 $4$.