Found problems: 387
Find all possible triples $(a, b, c)$ of positive integers with the following properties:
• $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$,
• $a$ is a divisor of $a + b + c$,
• $b$ is a divisor of $a + b + c$,
• $c$ is a divisor of $a + b + c$.
(Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.
For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$.
(Daniel Harrer)
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.
Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.
Prove that no convex $13$-gon can be cut into parallelograms.
Show that $n$ divides $2^n + 1$ for infinitely many positive integers $n$.
Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?
Given are $n$ integers. Prove that at least one of the following conditions applies:
1) One of the numbers is a multiple of $n$.
2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.
The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.
The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.
For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?
Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$
Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?
Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.
Among the $2n$ numbers $1, 2, 3, . . . , 2n$ are chosen in any way $n + 1$ different numbers. Prove that among the chosen numbers there are at least two, such that one divides the other.
Given odd prime $p$. Sequence ${x_n}$ is defined by $x_{n+2}= 4x_{n+1}-x_n$.
Choose $x_0,x_1$ such that for every random positive integer $k$, there exists $i\in \mathbb N$ such that $4p^2-8p+1|x_i - (2p)^k$.
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.
Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.
For $t \ge 2$, defi ne $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a [i]peak[/i] if $S(t) > S(u)$ for all values of $u < t$.
Prove or disprove the following statement:
For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.
Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties:
$\bullet$ $a+b$ divides $ab+1$,
$\bullet$ $a-b$ divides $ab -1$,
$\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.