5. The whole numbers from 1 to $2 k$ are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of $k$ ?
SolUtion
Numbers 30, 60 and 90 all share three distinct prime factors (2, 3 and 5). If $k \geq 45$, each of these three numbers must be assigned to one of the two groups, so one of these groups must contain at least two of these numbers. Hence $k88$, we have $x=p q r$ and $y=2 p q r$.
As $y \leq 88$ and $x=\frac{1}{2} y$, we have $x \leq 44$, so $x$ is in group $A$.
As $y=2 p q r \geq 2 \times(2 \times 3 \times 5)>44$, we have $y$ is in group $B$.
Hence 44 is the largest possible value of $k$.