Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.

Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$

Let $a,b $ be two relatively prime positive integers.Also let $m,n $ be positive integers with $n> m $.\\ Prove that\\ $lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}$ [i]Proposed by Navid Safaei[/i]

A whole number larger than $2$ leaves a remainder of $2$ when divided by each of the numbers $3, 4, 5$ and $6$. The smallest such number lies between which two numbers? $\textbf{(A)}\ 40\text{ and }49\qquad \textbf{(B)}\ 60\text{ and }79\qquad \textbf{(C)}\ 100\text{ and }129\qquad \textbf{(D)}\ 210\text{ and }249\qquad \textbf{(E)}\ 320\text{ and }369$

Let $a_1,b_1,c_1$ be natural numbers. We define \[a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\,\,b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\,\,c_2=\gcd(a_1,b_1),\] and \[a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\,\,b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\,\,c_3=\operatorname{lcm}(a_2,b_2).\] Show that $\gcd(b_3,c_3)=a_2$.

How many ordered pairs $\left(a,b\right)$ of positive integers are there such that \[\gcd\left(a,b\right)^3=\mathrm{lcm}\left(a,b\right)^2=4^6\] is true? [i]2019 CCA Math Bonanza Individual Round #4[/i]

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?

Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$

Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\cdots p_r^{i_r} (0 \le i_1,i_2,\cdots,i_r \le n).$ Alice and Bob play a game by making a move by turns, with Alice going first. In Alice's round, she erases two numbers $a,b$ (not necessarily different) and write $\gcd(a,b)$. In Bob's round, he erases two numbers $a,b$ (not necessarily different) and write $\mathrm{lcm} (a,b)$. The game ends when only one number remains on the blackboard. Determine the minimal possible $M$ such that Alice could guarantee the remaining number no greater than $M$, regardless of Bob's move.

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20 $

Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the $2$ upper books. Then he reverses the $3$ upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.

Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.

A set $S$ of positive integers is said to be [i]channeler[/i] if for any three distinct numbers $a,b,c \in S$, we have $a\mid bc$, $b\mid ca$, $c\mid ab$. a) Prove that for any finite set of positive integers $ \{ c_1, c_2, \ldots, c_n \} $ there exist infinitely many positive integers $k$, such that the set $ \{ kc_1, kc_2, \ldots, kc_n \} $ is a channeler set. b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.

Is there a natural number $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change.