This is where your prime factorisation skills come handy.

18 = 2 × 3 × 3 = 2 × 3²

We want rectangular array so we need to express 18 as a product of two factors.

One factor represents the length of the rectangular array (in terms of number of square tiles) , the other represents the breadth.


The different ways to write 18 as a product of two factors :


(2 × 3²) × 1 = 18 × 1
1 × (2 × 3²) = 1 × 18

We can either have length of 18 squares long, 1 square width or 18 squares width, 1 square length

Likewise,

(2 × 3) × 3 = 6 × 3
3 × (2 × 3) = 3 × 6

2 × 3² = 2 × 9
3² × 2 = 9 × 2

There are 6 ways in total to form the arrays.

Visually it will look like this :


⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛⬛
(1 × 18)

⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
⬛
(18 × 1)


⬛⬛⬛⬛⬛⬛⬛⬛⬛
⬛⬛⬛⬛⬛⬛⬛⬛⬛
(2 × 9)

⬛⬛
⬛⬛
⬛⬛
⬛⬛
⬛⬛
⬛⬛
⬛⬛
⬛⬛
⬛⬛
(9 × 2)


⬛⬛⬛⬛⬛⬛
⬛⬛⬛⬛⬛⬛
⬛⬛⬛⬛⬛⬛
(3 × 6)

⬛⬛⬛
⬛⬛⬛
⬛⬛⬛
⬛⬛⬛
⬛⬛⬛
⬛⬛⬛
(6 × 3)

But for b)

41 is a prime number.

It can only be expressed as a product of itself and 1.

41 × 1
1 × 41

So there are only 2 ways to form the array.

Thanks again for helping, you’re the best!