Found problems: 283
Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.
The smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6 lies between what numbers?
$\textbf{(A)}\hspace{.05in}40\text{ and }50 \qquad \textbf{(B)}\hspace{.05in}51\text{ and }55 \qquad \textbf{(C)}\hspace{.05in}56\text{ and }60 \qquad \textbf{(D)}\hspace{.05in} \text{61 and 65}\qquad \textbf{(E)}\hspace{.05in} \text{66 and 99}$
Determine if there exist five consecutive positive integers such that their LCM is a perfect square.
Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.
Does there exist a positive integer $m$, such that if $S_n$ denotes the lcm of $1,2, \ldots, n$, then $S_{m+1}=4S_m$?
$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.
Given sequence $\{a_n\}$ satisfying:
$$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$
It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.
a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$
b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$
Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers.
Prove that the series
$$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$
is convergent
Solve the equation
$mn =$ (gcd($m,n$))$^2$ + lcm($m, n$)
in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.
Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.
Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$
for all $x,y\in\mathbb N$.
Points $ A_1$, $ A_2$, $ ...$ are placed on a circle with center $ O$ such that $ \angle OA_n A_{n\plus{}1}\equal{}35^\circ$ and $ A_n\neq A_{n\plus{}2}$ for all positive integers $ n$. What is the smallest $ n>1$ for which $ A_n\equal{}A_1$?
Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number.
[i]Laurentiu Panaitopol[/i]
Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that
\[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]
Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$.
[i]Ray Li.[/i]
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
How many zeroes are there at the end the number $9^{999} + 1$?
Prove that for any $n \ge 1$,
$LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$
Given a positive integer $n$.
(a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$,
\[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\]
(b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.
Find all pairs $a, b$ of positive integers, for which
$$(a, b) + 3[a, b] = a^3 - b^3$$
Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$.
[i]Proposed by Oleksiy Masalitin[/i]