Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem B. 4698. (March 2015)

B. 4698. Give an example for sets $\displaystyle H_1,H_2,\ldots\subset\mathbb{N}$ for which the following conditions hold:

$\displaystyle a)$ $\displaystyle |H_n|=n$ for all positive integers $\displaystyle n$.

$\displaystyle b)$ For all positive integers $\displaystyle n$ and $\displaystyle k$, $\displaystyle H_n \cap H_k = H_{(n,k)}$, where $\displaystyle (n,k)$ is the greatest common divisor of $\displaystyle n$ and $\displaystyle k$.

(5 pont)

Deadline expired on April 10, 2015.

### Statistics:

 47 students sent a solution. 5 points: Baran Zsuzsanna, Bindics Boldizsár, Döbröntei Dávid Bence, Gáspár Attila, Glasznova Maja, Kerekes Anna, Kovács 246 Benedek, Lajkó Kálmán, Molnár-Sáska Zoltán, Nagy-György Pál, Porupsánszki István, Sal Kristóf, Schwarcz Tamás, Szebellédi Márton, Szécsényi Nándor, Tóth Viktor, Varga-Umbrich Eszter, Williams Kada, Zsakó Ágnes. 4 points: Andó Angelika, Árvai Balázs, Csépai András, Imolay András, Katona Dániel, Keresztfalvi Bálint, Mócsy Miklós, Nagy Dávid Paszkál, Nagy Kartal, Wei Cong Wu, Záhorský Ákos. 3 points: 4 students. 2 points: 6 students. 1 point: 2 students. 0 point: 3 students. Unfair, not evaluated: 2 solutions.

Problems in Mathematics of KöMaL, March 2015