Found problems: 310
We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.
Prove that there are in finitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.
Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.
Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.
Let $s(n)$ be the number of positive integer divisors of $n$.
Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.
For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors
$1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.
Find all primes $p$ such that $p^2 + 2543$ has at most $16$ divisors.
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.
Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$.
Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$.
Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds.
For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$
.
Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.
Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.
For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$
Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.
Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that
\[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]
Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.
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 $y$.)
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that
a) all primes are lonely,
b) there exist infinitely many non-lonely positive integers.