Show that it is possible to write a number from the set $\{1,2,3,4,5\}$ in each square of a $5\times120$ board ($5$ columns and $120$ rows) so that the following conditions are satisfied: (i) all numbers in the same row are distinct; (ii) all rows are distinct; (iii) the board can be partitioned into $24$ $5\times5$ boards which can be reassembled (without rotating and turning over) into a $120\times5$ board whose $120$ columns are distinct.