BALTIC-WAY 2016 Problem 2

Prove or disprove the following hypotheses.

a) For all $k \geq 2$, each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k$.

b) For all $k \geq 2$, each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.