Question 142
The $2014$ positive integers from 1 to 2014 are written on a blackboard. The allowed operation is to choose two numbers $a$ and $b$, erase them and write in their place the numbers $\gcd(a, b)$ (greatest common divisor) and $\operatorname{lcm}(a, b)$ (least common multiple). This operation can be performed with any two numbers on the board, including numbers that resulted from previous operations. Determine the largest number of 1s we can leave on the board.