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: 408

2014 Regional Olympiad of Mexico Center Zone, 1
Tags: number theory , divisible , divide , product
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

1968 Czech and Slovak Olympiad III A, 2
Tags: integer , number theory , divisible
Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.

2000 Chile National Olympiad, 5
Tags: power of 2 , divide , divisible , number theory
Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.

1949 Moscow Mathematical Olympiad, 156
Tags: number theory , divisible
Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.

1982 Poland - Second Round, 5
Tags: number theory , divide , divisible
Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

2013 Bosnia and Herzegovina Junior BMO TST, 1
Tags: number theory , sum of squares , divisible
It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

2017 QEDMO 15th, 9
Tags: number theory , divisible
Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.

1989 Tournament Of Towns, (205) 3
Tags: number theory , divisible , divide , digit
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

1911 Eotvos Mathematical Competition, 3
Tags: number theory , divide , divisible
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.

2019 Saudi Arabia BMO TST, 1
Tags: number theory , divide , divisible , prime
Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

2023 Assara - South Russian Girl's MO, 2
Tags: number theory , divisible , divide
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 $b$?

2021 Saudi Arabia BMO TST, 3
Tags: number theory , divide , divisible
Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

1977 Bundeswettbewerb Mathematik, 1
Tags: integer , algebra , number theory , sum , divisible
Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

1998 Tournament Of Towns, 1
Tags: number theory , divisible , divide
Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers? (Folklore)

2012 India Regional Mathematical Olympiad, 2
Tags: number theory , divide , divisible , power
Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

1932 Eotvos Mathematical Competition, 1
Tags: number theory , divisible
Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.

1947 Moscow Mathematical Olympiad, 137
Tags: number theory , divisible , combinatorics
a) $101$ numbers are selected from the set $1, 2, . . . , 200$. Prove that among the numbers selected there is a pair in which one number is divisible by the other. b) One number less than $16$, and $99$ other numbers are selected from the set $1, 2, . . . , 200$. Prove that among the selected numbers there are two such that one divides the other.

1954 Moscow Mathematical Olympiad, 267
Tags: divide , polynomial , divisible , algebra
Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

2017 May Olympiad, 5
Tags: number theory , divide , divisible
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.

2019 Austrian Junior Regional Competition, 4
Tags: number theory , divisible , divide , prime
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

1978 Chisinau City MO, 159
Tags: number theory , prime , divide , divisible , factorial
Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

2005 Thailand Mathematical Olympiad, 12
Tags: number theory , divide , divisible , even
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.

2011 Regional Olympiad of Mexico Center Zone, 4
Tags: number theory , digit , divisible
Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

2008 Estonia Team Selection Test, 4
Tags: sequence , number theory , divisible , consecutive
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

2003 Austrian-Polish Competition, 4
Tags: divide , divisible , product , number theory
A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.