A design is formed by arranging a certain number
of square tiles to form a rectangular array. Find
the number of possible distinct designs if there
are
(a) 18 
tiles
(b) 41 tiles
~~~~~~~~~~~~~~~~~~~
(a) Consider all possible different decompositions of the number 18 into the product
of two whole numbers
(1) 18 = 1*18
(2) 18 = 2*9
(3) 18 = 3*6
(4) 18 = 6*3
(5) 18 = 9*2
(6) 18 = 18*1
If you consider decompositions like (2) and (5) as different,
then the answer is "there are 6 different designs".
If you consider decompositions like (2) and (5) as identical,
then the answer is "there are 3 different designs".
(b) The same logic works in case (b).
In this case, the number 41 is a prime number, so, the two possible answers are
"there are two possible designs, if you consider the decompositions
41 = 1*41
and
41 = 41*1
as different"
or
"there is one possible design, if you consider the decompositions
41 = 1*41
and
41 = 41*1
as identical".
Solved.
