Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

Positive integers $p, q, r$ satisfy $gcd(a,b,c) = 1$. Prove that there exists an integer $a$ such that $gcd(p,q+ar) = 1$.

Prove that the equation \[ 6(6a^2 + 3b^2 + c^2) = 5n^2 \] has no solutions in integers except $a = b = c = n = 0$.

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.

Is it possible to arrange 1400 positive integer ( not necessarily distinct ) ,at least one of them being 2021 , around a circle such that any number on this circle equals to the sum of gcd of the two previous numbers and two next numbers? for example , if $a,b,c,d,e$ are five consecutive numbers on this circle , $c=\gcd(a,b)+\gcd(d,e)$

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.

Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils and $216$ erasers were distributed. This distribution was made so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. In this case, how many notebooks, pencils and erasers did each family receive?

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$

A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write \[\gcd\left(\dfrac ab,\dfrac cd\right) = \dfrac{\gcd(ad,bc)}{bd}.\] For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of positive rational numbers $(m,n)$ with $m<1$ and $n<1$ such that \[\gcd(m,n)=\dfrac{1}{K}.\] What is $f(2017)-f(2016)$?

Let $a_0=29$, $b_0=1$ and $$a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1}$$ for $n\geq 1$. Determine the smallest positive integer $k$ for which $29$ divides $\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does not divide $b_1$.

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

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 $gcd(n, n @m) = 1$.

Given a natural number $n{}$ find the smallest $\lambda$ such that\[\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda,\] for any positive integers $y{}$ and $x \geqslant y + n$.

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

Let $k>2$ be a positive integer. Elise and Xavier play a game that has four steps, in this order. [list=1] [*]Elise picks $2$ nonzero digits $(1-9)$, called $e$ and $f$. [*]Xavier then picks $k$ nonzero digits $(1-9)$, called $x_1,\cdots,x_k$. [*]Elise picks any positive integer $d$. [*]Xaiver picks an integer $b>10$.[/list] Each player's choices are known to the other player when the choices are made. The winner is determined as follows. Elise writes down the two-digit base $b$ number $ef_b$. Next, Xavier writes the $k$-digit base $b$ number that is constructed by concatenating his digits, \[(x_1\cdots x_k)_b.\] They then compute the greatest common divisor (gcd) of these two numbers. If this gcd is greater than or equal to the integer $d$ then Xavier wins. Otherwise Elise wins. (As an example game for $k=3$, Elise chooses the digits $(e, f) = (2, 4)$, Xavier chooses $(4, 4, 8)$, and then Elise picks $d = 100$. Xavier picks base $b = 25$. The base-25 numbers $2425$ and $44825$ are, respectively, equal to $54$ and $2608$. The greatest common divisor of these two is $2$, which is much less than $100$, so Elise wins handily.) Find all $k$ for which Xavier can force a win, no matter how Elise plays.

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.

Let $ABC$ be a triangle with integer sides in which $AB<AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB,AC)>1$.

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.