Found problems: 51
2008 Thailand Mathematical Olympiad, 7
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.
2013 Saudi Arabia BMO TST, 2
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 $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
2017 Purple Comet Problems, 4
Find the least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ and $b$.
2021 Bolivian Cono Sur TST, 2
Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$
2005 Cuba MO, 7
Determine all triples of positive integers $(x, y, z)$ that satisfy
$$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \
lcm(x, y, z) = 2400.$$
2021 Lotfi Zadeh Olympiad, 3
Find the least possible value for the fraction
$$\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}$$
over all distinct positive integers $a, b, c$.
By $lcm(x, y)$ we mean the least common multiple of $x, y$ and by $gcd(x, y)$ we mean the greatest common divisor of $x, y$.
2020 MMATHS, I5
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i]
2010 NZMOC Camp Selection Problems, 4
Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$ for
(i) $n = 2007$;
(ii) $n = 2010$.
1982 Kurschak Competition, 2
Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.
1978 Chisinau City MO, 156
The natural numbers $a_1 <a_2 <.... <a_n\le 2n$ are such that the least common multiple of any two of them is greater than $2n$. Prove that $a_1 >\left[\frac{2n}{3}\right]$.
2016 India PRMO, 15
Find the number of pairs of positive integers $(m; n)$, with $m \le n$, such that the ‘least common multiple’ (LCM) of $m$ and $n$ equals $600$.
2021 Durer Math Competition (First Round), 4
Determine all triples of positive integers $a, b, c$ that satisfy
a) $[a, b] + [a, c] + [b, c] = [a, b, c]$.
b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$.
Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.
2010 Brazil Team Selection Test, 2
Let $k > 1$ be a fixed integer. Prove that there are infinite positive integers $n$ such that
$$ lcm \, (n, n + 1, n + 2, ... , n + k) > lcm \, (n + 1, n + 2, n + 3,... , n + k + 1).$$
2021 Cyprus JBMO TST, 2
Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then
\[ \delta + \Delta = 4(\alpha + \beta) + 2021\]
2014 NZMOC Camp Selection Problems, 6
Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.
2015 Czech-Polish-Slovak Junior Match, 4
Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$.
2023 Azerbaijan JBMO TST, 1
Let $a < b < c < d < e$ be positive integers. Prove that
$$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$
where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?
2024 Indonesia MO, 8
Let $n \ge 2$ be a positive integer. Suppose $a_1, a_2, \dots, a_n$ are distinct integers. For $k = 1, 2, \dots, n$, let
\[ s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, \]
i.e. $s_k$ is the product of all terms of the form $|a_k - a_i|$, where $i \in \{ 1, 2, \dots, n \}$ and $i \not= k$.
Find the largest positive integer $M$ such that $M$ divides the least common multiple of $s_1, s_2, \dots, s_n$ for any choices of $a_1, a_2, \dots, a_n$.
1998 Tournament Of Towns, 1
(a) Prove that for any two positive integers a and b the equation $lcm (a, a + 5) = lcm (b, b + 5)$ implies $a = b$.
(b) Is it possible that $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$?
(A Shapovalov)
PS. part (a) for Juniors, both part for Seniors
2004 Switzerland - Final Round, 8
A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.
2022 Kyiv City MO Round 2, Problem 1
a) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+2d]$?
b) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+4d]$?
Here $[a, b]$ denotes the least common multiple of integers $a, b$.
2001 Singapore Team Selection Test, 3
Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$.
(i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$.
(ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.
2024 Czech and Slovak Olympiad III A, 1
Let $a, b, c$ be positive integers such that one of the values $$gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)$$
is equal to the product of the remaining two. Prove that one of the numbers $a, b, c$ is a multiple of another of them.
2011 Cuba MO, 7
Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.
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 $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.