Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6$$and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$
Problem
Source: Tuymaada 2019 Senior P5 out of 8
Tags: number theory, Fun, prime numbers
15.07.2019 13:00
This holds for any set of $9$ prime numbers $Q=\{p_1,p_2,\dots,p_9\}$. Take $a_1=p_1p_4p_3, a_2=p_1p_5p_6p_7, a_3=p_1p_2p_8,a_4=p_2p_5p_9, a_5=p_2p_3p_7$ and $a_6=p_3p_9p_8p_6$.
15.07.2019 13:14
Could u post the other problems? And is this for junior or senior
01.06.2020 19:42
XbenX wrote: $a_1=p_1p_4p_3, a_4=p_2p_5p_9$ These two are coprime .
01.06.2020 19:56
well if we take XbenX's solution and change $a_4=p_2p_4p_5p_9$ and $a_5 = p_2p_3p_6$ it might work @below typo
01.06.2020 20:06
JosefSvejk wrote: well if we take XbenX's solution and change $a_4=p_2p_4p_5p_9$ and $a_5 = p_1p_3p_6$ it might work @above $a_4$ and $a_5$ are co prime
01.06.2020 20:12
thanks for the solution