This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3

2005 Cuba MO, 7
Tags: lcm , gcd , gcd and lcm , number theory
Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

2022 May Olympiad, 4
Tags: number theory , gcd and lcm , combinatorics
a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

2024 Iberoamerican, 1
Tags: divisor , gcd and lcm , number theory
For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$. Prove that for all pairs of positive integers $(a,b)$ we have that: \[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \] and determine all pairs of positive integers $(a,b)$ where we have equality case.