Found problems: 583
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 $n$?
$\textbf{(A)}\ 13,860 \qquad
\textbf{(B)}\ 20,790 \qquad
\textbf{(C)}\ 21,560 \qquad
\textbf{(D)}\ 27,720 \qquad
\textbf{(E)}\ 41,580$
We call a positive integer $n{}$ [i]peculiar[/i] if, for any positive divisor $d{}$ of $n{}$ the integer $d(d + 1)$ divides $n(n + 1).$ Prove that for any four different peculiar positive integers $A, B, C$ and $D{}$ the following holds:
\[\gcd(A, B, C, D) = 1.\]
Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.
Let $ a, b, c$ be three natural numbers such that $ a < b < c$ and $ gcd (c \minus{} a, c \minus{} b) \equal{} 1$. Suppose there exists an integer $ d$ such that $ a \plus{} d, b \plus{} d, c \plus{} d$ form the sides of a right-angled triangle. Prove that there exist integers, $ l,m$ such that $ c \plus{} d \equal{} l^{2} \plus{} m^{2} .$
[b][Weightage 17/100][/b]
Let $C$ be a circle of radius $1$ and $O$ its center. Let $\overline{AB}$ be a chord of the circle and $D$ a point on $\overline{AB}$ such that $OD =\frac{\sqrt2}{2}$ such that $D$ is closer to $ A$ than it is to $ B$, and if the perpendicular line at $D$ with respect to $\overline{AB}$ intersects the circle at $E $and $F$, $AD = DE$. The area of the region of the circle enclosed by $\overline{AD}$, $\overline{DE}$, and the minor arc $AE$ may be expressed as $\frac{a + b\sqrt{c} + d\pi}{e}$ where $a, b, c, d, e$ are integers, gcd $(a, b, d, e) = 1$, and $c$ is squarefree. Find $a + b + c + d + e$
For any integer $n$ we define the numbers $a = n^5 + 6n^3 + 8n$ ¸ $b = n^4 + 4n^2 + 3$. Prove that the numbers $a$ and $b$ are relatively prime or have the greatest common factor of $3$.
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 , $c=\gcd(a,b)+\gcd(d,e)$
The sequence $\{a_n\}_{n\geq 1}$ is defined by $a_{n+2}=7a_{n+1}-a_n$ for positive integers $n$ with initial values $a_1=1$ and $a_2=8$. Another sequence, $\{b_n\}$, is defined by the rule $b_{n+2}=3b_{n+1}-b_n$ for positive integers $n$ together with the values $b_1=1$ and $b_2=2$. Find $\gcd(a_{5000},b_{501})$.
Let $ m,n$ be integers such that $ 0\le m\le 2n$. Then prove that the number $ 2^{2n \plus{} 2} \plus{} 2^{m \plus{} 2} \plus{} 1$ is perfect square iff $ m \equal{} n$.
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]
Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.
Let $a_{i}$, $i = 1,2, \dots ,n$, $n \geq 3$, be positive integers, having the greatest common divisor 1, such that \[a_{j}\textrm{ divide }\sum_{i = 1}^{n}a_{i}\]
for all $j = 1,2, \dots ,n$. Prove that \[\prod_{i = 1}^{n}a_{i}\textrm{ divides }\Big{(}\sum_{i = 1}^{n}a_{i}\Big{)}^{n-2}.\]
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 $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.
What is the greatest common factor of $12345678987654321$ and $12345654321$?
[i]Proposed by Evan Chen[/i]
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.
The symbols $ (a,b,\ldots,g)$ and $ [a,b,\ldots,g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $ a,b,\ldots,g$. For example, $ (3,6,18)\equal{}3$ and $ [6,15]\equal{}30$. Prove that \[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}\equal{}\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]
For any two positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}$$
As always, $[x,y]$ means the least common multiply of $x,y$.
[I]Proposed by A. Golovanov[/i]
What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None of above}
$
Find all sets $S$ of positive integers that satisfy all of the following.
$1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$.
$2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$.
$3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.
Let $A = (0, 0, 0)$ in 3D space. Define the [i]weight[/i] of a point as the sum of the absolute values of the coordinates. Call a point a [i]primitive lattice point[/i] if all of its coordinates are integers whose gcd is 1. Let square $ABCD$ be an [i]unbalanced primitive integer square[/i] if it has integer side length and also, $B$ and $D$ are primitive lattice points with different weights. Prove that there are infinitely many unbalanced primitive integer squares such that the planes containing the squares are not parallel to each other.
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$