Found problems: 310
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$.
($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.
Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$
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.
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.
A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.
Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.
Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.
Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$
Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$
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.
In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares
Find all natural numbers $n> 1$ for which the following applies:
The sum of the number $n$ and its second largest divisor is $2013$.
(R. Henner, Vienna)
$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$.
Prove that $m$ is a divisor of $n$.
Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?
A positive integer \( n \) satisfies the following conditions:
[list]
[*] The number \( n \) has exactly \( 60 \) divisors: \( 1 = a_1 < a_2 < \cdots < a_{60} = n \);
[*] The number \( n+1 \) also has exactly \( 60 \) divisors: \( 1 = b_1 < b_2 < \cdots < b_{60} = n+1 \).
[/list]
Let \( k \) be the number of indices \( i \) such that \( a_i < b_i \). Find all possible values of \( k \).
[i]Note: Such numbers exist, for example, the numbers \( 4388175 \) and \( 4388176 \) both have \( 60 \) divisors.[/i]
[i]Proposed by Anton Trygub[/i]
Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.
For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$.
Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.
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.