For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

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]

Let $n$ be a positive integer. Let $S$ be a set of $n$ positive integers such that the greatest common divisors of all nonempty sets of $S$ are distinct. Determine the smallest possible number of distinct prime divisors of the product of the elements of $S$.

Let $S \subset \{ 1, \dots, n \}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$. We assume that $s \not= 1$ and let $d$ be its smallest divisor greater than $1$. Let $T \subset \{ 1, \dots, n \}$ be a set such that $S \subset T$ and $|T| \ge 1 + \left[ \frac{n}{d} \right]$. Prove that the greatest common divisor of the elements in $T$ is $1$. ----------- [Second Version] Let $n(n \ge 1)$ be a positive integer and $U = \{ 1, \dots, n \}$. Let $S$ be a nonempty subset of $U$ and let $d (d \not= 1)$ be the smallest common divisor of all elements of the set $S$. Find the smallest positive integer $k$ such that for any subset $T$ of $U$, consisting of $k$ elements, with $S \subset T$, the greatest common divisor of all elements of $T$ is equal to $1$.

Let $c$ be the least common multiple of positive integers $a$ and $b$, and $d$ be the greatest common divisor of $a$ and $b$. How many pairs of positive integers $(a,b)$ are there such that \[ \dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1? \] $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 2 $

Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

The greatest integer that will divide $13,511$, $13,903$, and $14,589$ and leave the same remainder is $\textbf{(A) }28\qquad\textbf{(B) }49\qquad\textbf{(C) }98\qquad$ $\textbf{(D) }\text{an odd multiple of }7\text{ greater than }49\qquad \textbf{(E) }\text{an even multiple of }7\text{ greater than }98$

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]

Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.

For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!\plus{}1$ and $ (n\plus{}1)!$. Find, without proof, a formula for $ f(n)$.

For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$ (a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$. (b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?

Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]

Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

Let $m, n$ be integers such that $n \geq m \geq 1$. Prove that $\frac{\text{gcd} (m,n)}{n} \binom{n}{m}$ is an integer. Here $\text{gcd}$ denotes greatest common divisor and $\binom{n}{m} = \frac{n!}{m!(n-m)!}$ denotes the binomial coefficient.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.

Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

There are exactly $77,000$ ordered quadruples $(a,b,c,d)$ such that $\gcd(a,b,c,d)=77$ and $\operatorname{lcm}(a,b,c,d)=n$. What is the smallest possible value of $n$? $\textbf{(A)}\ 13,860 \qquad \textbf{(B)}\ 20,790 \qquad \textbf{(C)}\ 21,560 \qquad \textbf{(D)}\ 27,720 \qquad \textbf{(E)}\ 41,580$

For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.