Found problems: 283
2008 Postal Coaching, 1
Prove that for any $n \ge 1$,
$LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$
2012 Online Math Open Problems, 4
Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$.
[i]Ray Li.[/i]
2006 AMC 12/AHSME, 19
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
1958 AMC 12/AHSME, 32
With $ \$1000$ a rancher is to buy steers at $ \$25$ each and cows at $ \$26$ each. If the number of steers $ s$ and the number of cows $ c$ are both positive integers, then:
$ \textbf{(A)}\ \text{this problem has no solution}\qquad\\
\textbf{(B)}\ \text{there are two solutions with }{s}\text{ exceeding }{c}\qquad \\
\textbf{(C)}\ \text{there are two solutions with }{c}\text{ exceeding }{s}\qquad \\
\textbf{(D)}\ \text{there is one solution with }{s}\text{ exceeding }{c}\qquad \\
\textbf{(E)}\ \text{there is one solution with }{c}\text{ exceeding }{s}$
2014 District Olympiad, 4
Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.
2020 Latvia Baltic Way TST, 16
Given sequence $\{a_n\}$ satisfying:
$$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$
It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.
1990 Vietnam Team Selection Test, 1
Let $ T$ be a finite set of positive integers, satisfying the following conditions:
1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$.
2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$.
For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.
2007 JBMO Shortlist, 1
Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ ,
where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.
2016 AMC 12/AHSME, 22
How many ordered triples $(x, y, z)$ of positive integers satisfy $\text{lcm}(x, y) = 72$, $\text{lcm}(x, z)= 600$, and $\text{lcm}(y, z) = 900$?
$\textbf{(A) } 15 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 64$
2000 Junior Balkan Team Selection Tests - Moldova, 2
The number $665$ is represented as a sum of $18$ natural numbers nenule $a_1, a_2, ..., a_{18}$.
Determine the smallest possible value of the smallest common multiple of the numbers $a_1, a_2, ..., a_{18}$.
2020 Tournament Of Towns, 2
Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.)
Boris Frenkin
1984 IMO Longlists, 7
Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.
PEN A Problems, 68
Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]
2018 AMC 10, 23
How many ordered pairs $(a, b)$ of positive integers satisfy the equation
$$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$
2014 Cono Sur Olympiad, 1
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 $2014$ $\times$ $\sqrt[2014]{2014!}$
2004 AMC 10, 4
A standard six-sided die is rolled, and $ P$ is the product of the fi ve numbers that are visible. What is the largest number that is certain to divide $ P$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 144\qquad
\textbf{(E)}\ 720$
2015 Indonesia MO Shortlist, N7
For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies
\[
4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}
\]
PEN H Problems, 86
A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?
2016 AMC 8, 20
The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?
$\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$
2004 Postal Coaching, 2
(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$
(b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.
2014 Contests, 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 $\frac{a+b}{4}$
1972 USAMO, 1
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 \[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}\equal{}\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]
2020 Estonia Team Selection Test, 1
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$.
Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .
2014 Switzerland - Final Round, 2
Let $a,b\in\mathbb{N}$ such that :
\[ ab(a-b)\mid a^3+b^3+ab \]
Then show that $\operatorname{lcm}(a,b)$ is a perfect square.
2010 AMC 8, 20
In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20 $