IT-CESENATICO 2011 Problem 2

A sequence of positive integers $a_1, a_2,\ldots, a_n$ is called ladder of length $n$ if it consists of $n$ consecutive integers in ascending order. (a) Prove that for every positive integer $n$ there exist two ladders 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 all $i$ between $1$ and $n$ , the greatest common divisor of $a_i$ and $b_i$ is equal to $1$ . (b) Prove that for every positive integer $n$ there exist two ladders 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 all $i$ between $1$ and $n$ , the greatest common divisor of $a_i$ and $b_i$ is greater than $1$ .