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

1999 Singapore MO Open, 2
Tags: number theory , digit , divisible , divide
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

2024 Regional Competition For Advanced Students, 4
Tags: number theory , divisible , divide
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

1947 Kurschak Competition, 1
Tags: number theory , divide , divisible
Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

2010 QEDMO 7th, 9
Tags: number theory , divisible
Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that $$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.

2001 Austria Beginners' Competition, 1
Tags: number theory , divide , divisible
Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

1975 Chisinau City MO, 101
Tags: number theory , divisible , difference
Prove that among any $k + 1$ natural numbers there are two numbers whose difference is divisible by $k$.

2010 Saudi Arabia BMO TST, 3
Tags: number theory , divisible , divide
How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

1990 ITAMO, 5
Tags: number theory , divisible , trinomial
Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.

2016 Saudi Arabia Pre-TST, 2.4
Tags: number theory , product , divide , divisible
Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

2013 Costa Rica - Final Round, 6
Tags: number theory , digit , divisible
Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.

1975 Chisinau City MO, 87
Tags: number theory , divisible , difference
Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.

1999 Tournament Of Towns, 1
Tags: combinatorics , divisible , divide , sum
For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

2017 Saudi Arabia BMO TST, 1
Tags: number theory , divide , divisible , factorial
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

2011 Hanoi Open Mathematics Competitions, 7
Tags: number theory , divisible
How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.

1952 Moscow Mathematical Olympiad, 226
Tags: number theory , divisible
Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.

2012 Mathcenter Contest + Longlist, 3
Tags: divisible , prime , number theory
If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

2015 Belarus Team Selection Test, 1
Tags: number theory , divisible , consecutive , prime
Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

2002 Estonia National Olympiad, 2
Tags: digit , divisible , number theory
Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?

2011 Hanoi Open Mathematics Competitions, 1
Tags: number theory , divisible
An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$. How many integers between $1$ and $100$ are octal? (A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$

1940 Moscow Mathematical Olympiad, 065
Tags: divisible , number theory
How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?

2013 Saudi Arabia Pre-TST, 2.1
Tags: number theory , remainder , divide , divisible
Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

2020 Israel Olympic Revenge, N
Tags: number theory , divisible
Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers. Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.

1970 All Soviet Union Mathematical Olympiad, 137
Tags: number theory , divisible , sum , combinatorics
Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

VMEO IV 2015, 10.3
Tags: number theory , divide , divisible
Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

1999 Abels Math Contest (Norwegian MO), 2b
Tags: divide , divisible , number theory
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$