Question 45
(a) Prove that for every positive integer $n$ there exist two “staircases” of length $n$, with no elements in common, $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$, such that for every $i$ from $1$ to $n$ the greatest common divisor of $a_i$ and $b_i$ is $1$.
(b) Prove that for every positive integer $n$ there exist two “staircases” of length $n$, with no elements in common, $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$, such that for every $i$ from $1$ to $n$ the greatest common divisor of $a_i$ and $b_i$ is greater than $1$.
(b) Prove that for every positive integer $n$ there exist two “staircases” of length $n$, with no elements in common, $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$, such that for every $i$ from $1$ to $n$ the greatest common divisor of $a_i$ and $b_i$ is greater than $1$.