Found problems: 283
1959 AMC 12/AHSME, 42
Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. Then, which two of the following statements are true?
$ \text{(1)}\ \text{the product MD cannot be less than abc} \qquad$
$\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad$
$\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad$
$\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}$ $\text{ (This means: no two have a common factor greater than 1.)}$
$ \textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $
2010 Czech And Slovak Olympiad III A, 6
Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.
2006 AMC 8, 23
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 5$
1994 AMC 8, 1
Which of the following is the largest?
$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{3}{8} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{7}{24}$
2016 AMC 10, 25
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$
1990 Federal Competition For Advanced Students, P2, 1
Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime.
2004 Iran MO (3rd Round), 18
Prove that for any $ n$, there is a subset $ \{a_1,\dots,a_n\}$ of $ \mathbb N$ such that for each subset $ S$ of $ \{1,\dots,n\}$, $ \sum_{i\in S}a_i$ has the same set of prime divisors.
PEN A Problems, 15
Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.
2018 SIMO, Q1
Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$
2010 Contests, 3
The sum $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
2014 India National Olympiad, 3
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 $\frac{a+b}{4}$
2018 China Western Mathematical Olympiad, 7
Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that
$$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$
2004 Dutch Mathematical Olympiad, 1
Determine the number of pairs of positive integers $(a, b)$, with $a \le b$, for which lcm $(a, b) = 2004$.
lcm ($a, b$) means the least common multiple of $a$ and $b$. Example: lcm $(18, 24) = 72$.
2019 CCA Math Bonanza, I4
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]
2005 Bosnia and Herzegovina Team Selection Test, 3
Let $n$ be a positive integer such that $n \geq 2$. Let $x_1, x_2,..., x_n$ be $n$ distinct positive integers and $S_i$ sum of all numbers between them except $x_i$ for $i=1,2,...,n$. Let $f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}.$
Determine maximal value of $f(x_1,x_2,...,x_n)$, while $(x_1,x_2,...,x_n)$ is an element of set which consists from all $n$-tuples of distinct positive integers.
2022 AMC 10, 19
Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots +\frac{1}{17}=\frac{h}{L_{17}}$. What is the remainder when $h$ is divided by $17?$
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
1993 Greece National Olympiad, 6
What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?
1951 Miklós Schweitzer, 8
Given a positive integer $ n>3$, prove that the least common multiple of the products $ x_1x_2\cdots x_k$ ($ k\geq 1$) whose factors $ x_i$ are positive integers with $ x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n$, is less than $ n!$.
1951 Moscow Mathematical Olympiad, 204
* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.
2021 Francophone Mathematical Olympiad, 4
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 $b$.
2022 Kyiv City MO Round 2, Problem 1
Find all triples $(a, b, c)$ of positive integers for which $a + [a, b] = b + [b, c] = c + [c, a]$.
Here $[a, b]$ denotes the least common multiple of integers $a, b$.
[i](Proposed by Mykhailo Shtandenko)[/i]
2020 China Team Selection Test, 3
For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$
Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.
2003 Iran MO (3rd Round), 8
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
1995 Tournament Of Towns, (464) 2
Do there exist $100$ positive integers such that their sum is equal to their least common multiple?
(S Tokarev)
2021 Czech-Polish-Slovak Junior Match, 3
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 $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.