### Divisibility & Induction

Food for thought

Like a standard Sudoku, this Sudoku variant has the basic rule that every row, every column and every $3 \times 3$ box in the grid contains the digits $1$ to $9$.
The first step to solving this puzzle is to find the values of the unknown digits (all indicated by asterisks) in the cells of the $9 \times 9$ grid. At the bottom and right side of the $9 \times 9$ grid are numbers, each of which is the least common multiple (LCM) of all the starred numbers in the row or column preceding it.
We define the least common multiple of a set of whole numbers as the smallest whole number which is divisible by all of them. For example, the least common multiple of $3$, $4$ and $8$ is $24$.