Found problems: 583
Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.
For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$
Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \[\gcd(n,2^n - 1) = 3.\]
Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that
$$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$
where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$
Find the greatest common divisor of all number in the set $( a^{41} - a | a \in \mathbb{N} and \geq 2 )$ . What is your guess if 41 is replaced by a natural number $n$
Find all possible pairs of integers $ a$ and $ b$ such that $ab = 160 + 90 (a,b)$, where $(a, b)$ is the greatest common divisor of $ a$ and $ b$.
Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$
[i]Proposed by Daniel Liu[/i]
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$
The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.
Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.
Prove there exist positive integers a,b,c for which $a+b+c=1998$, the gcd is maximized, and $0<a<b\leq c<2a$.
Find those numbers.
Are they unique?
For any positive integers $a,b,c,n$, we define
$$D_n(a,b,c)=\mathrm{gcd}(a+b+c,a^2+b^2+c^2,a^n+b^n+c^n).$$
1. Prove that if $n$ is a positive integer not divisible by $3$, then for any positive integer $k$, there exist three integers $a,b,c$ such that $\mathrm{gcd}(a,b,c)=1$, and $D_n(a,b,c)>k$.
2. For any positive integer $n$ divisible by $3$, find all values of $D_n(a,b,c)$, where $a,b,c$ are three positive integers such that $\mathrm{gcd}(a,b,c)=1$.
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.
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 $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{pmatrix}$$
let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$
Prove that the following statements are equivalent:
$\textbf{i) }$ $|\det A | = 1$
$\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$
[i]Romeo Raicu[/i]
A set $S$ of positive integers is called magic if for any two distinct members of $S, i$ and $j$, $\frac{i+ j}{GCD(i, j)}$is also a member of $S$.
The $GCD$, or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, $GCD(36,80) = 4$.
Find and describe all finite magic sets.
Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.
For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$. Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained.
The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$
Suppose that $a$ and $b$ are two distinct positive real numbers such that $\lfloor na\rfloor$ divides $\lfloor nb\rfloor$ for any positive integer $n$. Prove that $a$ and $b$ are positive integers.
Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?
Find all possible triples $(a, b, c)$ of positive integers with the following properties:
• $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$,
• $a$ is a divisor of $a + b + c$,
• $b$ is a divisor of $a + b + c$,
• $c$ is a divisor of $a + b + c$.
(Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)