Discussion
Ammini is arranging 36 identical squares in the form of different rectangles. How many different types of rectangles can she make with these squares ?
Answer: C
She can make five different rectangles.
Each rectangle made from 36 unit squares has sides whose product is 36. List the factor pairs, counting each shape once whichever way round it is placed:
\[ \begin{aligned} 36 &= 1 \times 36 = 2 \times 18 = 3 \times 12 \\ &= 4 \times 9 = 6 \times 6 \end{aligned} \]
That is five arrangements (the 6 by 6 one is a square, which is a special rectangle). Eight comes from counting 36 x 1 and 1 x 36 as different, and four from leaving out the square.
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