Question 3
Alex begins by writing down the number $100$ on a piece of paper. Starting from the second number onward, each number written on the paper is the sum of the previous number and its largest prime factor. For example, the second number is $100+5=105$ since the largest prime factor of $100$ is $5$, and the third number is $105+7=112$ since the largest prime factor of $105$ is $7$. What is the smallest possible number written by Alex that is greater than or equal to $2026$?