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: 387

2013 Saudi Arabia Pre-TST, 3.2
Tags: divide , divisible , number theory
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.

1956 Moscow Mathematical Olympiad, 323
Tags: divide , divisor , number theory
a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

2016 Dutch BxMO TST, 5
Tags: divide , number theory
Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

2017 Junior Balkan Team Selection Tests - Romania, 3
Tags: divide , number theory
Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.

2021 Saudi Arabia Training Tests, 39
Tags: sum , divide , divisible , number theory
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

2020 Argentina National Olympiad, 1
Tags: sum of digits , divide , number theory
For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.

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

2002 Junior Balkan Team Selection Tests - Romania, 1
Tags: divide , divisor , sum of powers , number theory , divisible
Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

2013 Tournament of Towns, 6
Tags: prime , fraction , divide , divisor , prime number , number theory
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

2013 Saudi Arabia GMO TST, 3
Tags: divide , divisible , number theory
Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

2011 Tournament of Towns, 6
Tags: number theory , divide , divisor , power of 2
Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.

2013 Saudi Arabia Pre-TST, 4.2
Tags: number theory , divide , divisible
Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

1983 Tournament Of Towns, (038) A5
Tags: divide , number theory , divisible , combinatorics
Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)

2008 Mathcenter Contest, 6
Tags: divide , number theory
For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$. [i](tomoyo-jung)[/i]

1956 Polish MO Finals, 4
Tags: divide , divisible , number theory
Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

2006 Singapore Senior Math Olympiad, 1
Tags: mutliple , divide , divisible , prime , number theory
Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2013 Thailand Mathematical Olympiad, 5
Tags: number theory , sum of digits , digit , divide , divisible
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.

1997 Tournament Of Towns, (537) 2
Tags: number theory , divisible , divide
Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

2018 Czech-Polish-Slovak Junior Match, 1
Tags: divide , divisible , number theory
For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$. Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$

2017 May Olympiad, 5
Tags: divide , divisible , number theory
We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.

2013 Saudi Arabia BMO TST, 4
Tags: divide , number theory , divisible
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.

2007 Estonia National Olympiad, 4
Tags: number theory , greatest common divisor , divide
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third. a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient. b) Give an example of one of these larger numbers $a, b$ and $c$

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 $m$.

2021 Saudi Arabia JBMO TST, 3
Tags: number theory , divide , divisible
We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

2017 Auckland Mathematical Olympiad, 3
Tags: number theory , digit , divide , divisible
The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.