Estonian Mathematical Olympiad

Let $n$ be a positive integer and $d$ one of its positive divisors. Integers $1$ to $n$ are written in columns of length $d$ (the first column contains the numbers $1$ to $d$, starting from the top, the second column contains the numbers $d+1$ to $2d$ etc.). Then, one finds the greatest common divisor of each row and finally the least common multiple of the greatest common divisors. Prove that if $d<n$, then the final result is equal to $d$.