Found problems: 283
2013 Canadian Mathematical Olympiad Qualification Repechage, 3
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.
2009 Saint Petersburg Mathematical Olympiad, 2
$[x,y]-[x,z]=y-z$ and $x \neq y \neq z \neq x$
Prove, that $x|y,x|z$
2013 Switzerland - Final Round, 1
Find all triples $(a, b, c)$ of natural numbers such that the sets
$$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and
$$\{2, 3, 5, 30, 60\}$$
are the same.
Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.
2008 Putnam, A3
Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)
2016 Dutch IMO TST, 3
Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.
2010 Tournament Of Towns, 5
$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?
2010 Purple Comet Problems, 9
What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$
2010 Korea - Final Round, 6
An arbitrary prime $ p$ is given. If an integer sequence $ (n_1 , n_2 , \cdots , n_k )$ satisfying the conditions
- For all $ i\equal{} 1, 2, \cdots , k$, $ n_i \geq \frac{p\plus{}1}{2}$
- For all $ i\equal{} 1, 2, \cdots , k$, $ p^{n_i} \minus{} 1$ is divisible by $ n_{i\plus{}1}$, and $ \frac{p^{n_i} \minus{} 1}{n_{i\plus{}1}}$ is coprime to $ n_{i\plus{}1}$. Let $ n_{k\plus{}1} \equal{} n_1$.
exists not for $ k\equal{}1$, but exists for some $ k \geq 2$, then call the prime a good prime.
Prove that a prime is good iff it is not $ 2$.
2014 Online Math Open Problems, 6
Let $L_n$ be the least common multiple of the integers $1,2,\dots,n$. For example, $L_{10} = 2{,}520$ and $L_{30} = 2{,}329{,}089{,}562{,}800$. Find the remainder when $L_{31}$ is divided by $100{,}000$.
[i]Proposed by Evan Chen[/i]
2007 Tournament Of Towns, 1
[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.
[b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
2017 CMIMC Number Theory, 2
Determine all possible values of $m+n$, where $m$ and $n$ are positive integers satisfying \[\operatorname{lcm}(m,n) - \gcd(m,n) = 103.\]
2015 Junior Balkan Team Selection Tests - Moldova, 8
Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities:
$$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$
([x, y]represents the least common multiple of positive integers x,y)
2008 Spain Mathematical Olympiad, 1
Find two positive integers $a$ and $b$, when their sum and their least common multiple is given. Find the numbers when the sum is $3972$ and the least common multiple is $985928$.
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 $i = 0, 1, 2$.
2011 Saint Petersburg Mathematical Olympiad, 2
$a,b$ are naturals and $$a \times GCD(a,b)+b \times LCM(a,b)<2.5 ab$$. Prove that $b|a$
PEN N Problems, 9
Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that
a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$,
b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$.
Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.
2004 All-Russian Olympiad, 4
Is there a natural number $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change.
1995 USAMO, 4
Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:
(i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$
(ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$
Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.
2013 India Regional Mathematical Olympiad, 5
Let $a_1,b_1,c_1$ be natural numbers. We define \[a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\,\,b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\,\,c_2=\gcd(a_1,b_1),\] and \[a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\,\,b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\,\,c_3=\operatorname{lcm}(a_2,b_2).\] Show that $\gcd(b_3,c_3)=a_2$.
1981 IMO, 1
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?
[b]b.)[/b] For which $n>2$ is there exactly one set having this property?
2011 Singapore Senior Math Olympiad, 2
Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.
1951 AMC 12/AHSME, 37
A number which when divided by $ 10$ leaves a remainder of $ 9$, when divided by $ 9$ leaves a remainder of $ 8$, by $ 8$ leaves a remainder of $ 7$, etc., down to where, when divided by $ 2$, it leaves a remainder of $ 1$, is:
$ \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 419 \qquad\textbf{(C)}\ 1259 \qquad\textbf{(D)}\ 2519 \qquad\textbf{(E)}\ \text{none of these answers}$
2014 Contests, 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 $2014$ $\times$ $\sqrt[2014]{2014!}$
1974 IMO Shortlist, 7
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
2023 AMC 12/AHSME, 24
Integers $a, b, c, d$ satisfy the following:
$abcd=2^6\cdot 3^9\cdot 5^7$
$\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$
$\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$
$\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$
$\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$
Find $\text{gcd}(a,b,c,d)$
$\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$