Found problems: 310
Determine all pairs $(a, b)$ of integers having the following property:
there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.
Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.
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 $n$.
A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?
Every positive integer is either [i]nice [/i] or [i]naughty[/i], and the Oracle of Numbers knows which are which.
However, the Oracle will not directly tell you whether a number is [i]nice [/i] or [i]naughty[/i]. The only questions the Oracle will answer are questions of the form “What is the sum of all nice divisors of $n$?,” where $n$ is a number of the questioner’s choice. For instance, suppose ([i]just [/i] for this example) that $2$ and $3$ are nice, while $1$ and $6$ are [i]naughty[/i]. In that case, if you asked the Oracle, “What is the sum of all nice divisors of $6$?,” the Oracle’s answer would be $5$.
Show that for any given positive integer $n$ less than $1$ million, you can determine whether $n$ is [i]nice [/i] or [i]naughty [/i] by asking the Oracle at most four questions.
For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$.
For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?
Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.
For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.
Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$?
[i]Proposed by Pavel Ciurea[/i]
Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?
A sequence of natural numbers is written according to the following rule:
[i] the first two numbers are chosen and thereafter, in order to write a new number, the sum of the last numbers is calculated using the two written numbers, we find the greatest odd divisor of their sum and the sum of this greatest odd divisor plus one is the following written number.
[/i]The first numbers are $25$ and $126$ (in that order), and the sequence has $2015$ numbers. Find the last number written.
Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied?
$\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above.
$\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).
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]
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.
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 $n$.
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$).
Prove that there are infi nite number of [i]Primes of the Year[/i].
For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that:
(a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$.
(b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.
Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$).
(Tomáš Bárta)
Given a positive integer $n$, let $D$ is the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent:
(a) For any positive divisor $m$ of $n$,
\[ n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. \]
(b) For any positive divisor $k$ of $n$,
\[ k ~\Big|~ \sum_{d|k} f(d). \]
On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $18$. We know that the sum of all numbers on the paper having exactly one proper divisor is $666$. What is the sum of all numbers on the paper having exactly two proper divisors?
We say that $k$ is a [i]proper divisor [/i]of the positive integer $n$ if $k | n$ and $1 < k < n$.
We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.
If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$
Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.
We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?