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39

Part of PEN O Problems

Problems(1)

O 39

Source:

5/25/2007
Find the smallest positive integer nn for which there exist nn different positive integers a1,a2,⋯ ,ana_{1}, a_{2}, \cdots, a_{n} satisfying [*] lcm(a1,a2,⋯ ,an)=1985\text{lcm}(a_1,a_2,\cdots,a_n)=1985,[*] for each i,j∈{1,2,⋯ ,n}i, j \in \{1, 2, \cdots, n \}, gcd(ai,aj)≠1gcd(a_i,a_j)\not=1, [*] the product a1a2⋯ana_{1}a_{2} \cdots a_{n} is a perfect square and is divisible by 243243, and find all such nn-tuples (a1,⋯ ,an)(a_{1}, \cdots, a_{n}).
number theoryleast common multiplegreatest common divisorprime factorization