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: 408

2007 QEDMO 4th, 4
Tags: number theory , divisible
Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

1977 Swedish Mathematical Competition, 1
Tags: number theory , divide , divisible
$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.

2008 Switzerland - Final Round, 3
Tags: number theory , divide , divisible
Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.

1997 All-Russian Olympiad Regional Round, 10.3
Tags: number theory , divisible , divide
Natural numbers $m$ and $n$ are given. Prove that the number $2^n-1$ is divisible by the number $(2^m -1)^2$ if and only if the number $n$ is divisible by the number $m(2^m-1)$.

2009 Thailand Mathematical Olympiad, 10
Tags: number theory , divide , divisible
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.

2009 Hanoi Open Mathematics Competitions, 5
Tags: number theory , divisible
Prove that $m^7- m$ is divisible by $42$ for any positive integer $m$.

1987 ITAMO, 1
Tags: number theory , divisible
Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$

2018 Dutch Mathematical Olympiad, 1
Tags: number theory , digit , divisible
We call a positive integer a [i]shuffle[/i] number if the following hold: (1) All digits are nonzero. (2) The number is divisible by $11$. (3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$. How many $10$-digit [i]shuffle[/i] numbers are there?

2015 Caucasus Mathematical Olympiad, 1
Tags: digit , number theory , divisible , sum
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

1938 Moscow Mathematical Olympiad, 042
Tags: number theory , divisible , combinatorics
How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?

2010 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible
Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

2014 Saudi Arabia Pre-TST, 4.1
Tags: number theory , inequalities , divide , divisible
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

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)

2021 Auckland Mathematical Olympiad, 4
Tags: power , number theory , divide , divisible
Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

2007 Bosnia and Herzegovina Junior BMO TST, 2
Tags: number theory , divisible , divide
Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

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 Poland - Second Round, 4
Tags: number theory , divide , divisible
Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.

1965 Czech and Slovak Olympiad III A, 1
Tags: number theory , divisible , divisibility tests
Show that the number $5^{2n+1}2^{n+2}+3^{n+2}2^{2n+1}$ is divisible by $19$ for every non-negative integer $n$.

2015 Caucasus Mathematical Olympiad, 1
Tags: number theory , divisible , remainder , digit
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

Mathley 2014-15, 8
Tags: number theory , divisible , power of 2 , divide
For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$ where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$. Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi

2016 Saudi Arabia Pre-TST, 1.4
Tags: number theory , coprime , divide , divisible
Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$. 1. Show that $F(r)$ is a positive integer for any prime $r \ne p$. 2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$. 3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.

1970 Polish MO Finals, 3
Tags: binomial coefficient , number theory , divisible , prime
Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.

VII Soros Olympiad 2000 - 01, 8.3
Tags: number theory , divisible , algebra
Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

2018 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible , functional
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

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)