Found problems: 310
Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?
Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number
\[
\frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}}
\]
is an integer?
An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself.
For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$.
Let $n!$ denote the product $1\cdot 2\cdot 3\cdot ...\cdot n$, where $n$ is a positive integer.
An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$.
For example, $24$ is a factorial number since $24 = 4! = 1\cdot 2\cdot 3\cdot 4$.
Find all perfect numbers greater than $1$ that are also factorials.
Let $n>4$ be a positive integer, which is divisible by $4$. We denote by $A_n$ the sum of the odd positive divisors of $n$. We also denote $B_n$ the sum of the even positive divisors of $n$, excluding the number $n$ itself. Find the least possible value of the expression $$f(n)=B_n-2A_n,$$
for all possible values of $n$, as well as for which positive integers $n$ this minimum value is attained.
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if
\[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.
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 a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.
Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.
Positive integer $n$ is such that number $n^2-9$ has exactly $6$ positive divisors. Prove that GCD $(n-3, n+3)=1$
Max divided a natural number $p$ by a natural number $q \leq 100$. In the decimal representation of the quotient he calculated, the sequence of digits $1982$ occurs somewhere after the decimal point. Show that Max made a computational mistake.
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
$\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$
Given a positive integer $n$. The number $2n$ has $28$ positive factors, while
the number $3n$ has $30$ positive factors.
Find the number of positive divisors of $6n$.
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.
Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and
a) Product of all divisor of $sm, sn$ are equal.
b) Number of divisors of $sm,sn$ are equal.
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $
Let $s,t$ be two different positive integers with the following property:
If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$.
Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer.
(FYROM)
It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$
For an integer $a \ge 2$, denote by $\delta_(a) $ the second largest divisor of $a$. Let $(a_n)_{n\ge 1}$ be a sequence
of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$
for all $n \ge 1$. Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$.
Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$.
a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$.
b) Find the number of all non-negative solutions of the above equation.
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.
Given a natural number $n$, call a divisor $d$ of $n$ to be $\textit{nontrivial}$ if $d>1$. A natural number $n$ is $\textit{good}$ if one or more distinct nontrivial divisors of $n$ sum up to $n-1$.
Prove that every natural number $n$ has a multiple that is good.
We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.
For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.
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 a natural number $n$, let $D (n)$ be the set of (positive integers) divisors of $n$. Furthermore let $d (n)$ be the number of divisors of $n,$ that is, the cardinality of $D (n)$. For each such $n$, prove the equality $$\sum_{k\in D(n)} d(k)^3=\left( \sum_{k\in D(n)} d(k)\right) ^2.$$
a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$.
b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.