Problem 15 · 2016 Math Kangaroo
Hard
Number Theory
factor-pairs
Diana cuts a rectangle of area 2016 into 56 identical squares. The side lengths of the rectangle and the squares are all whole numbers. For how many different rectangles can she do this? (Two rectangles are said to be different if they are not congruent.)
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Answer: B — 4
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Hint 1 of 2
Each square has the same area, so find that area first.
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Hint 2 of 2
2016/56 = 36, so squares are 6x6; count the rectangle shapes from the factor pairs of 56.
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Approach: fix the square size, count factor pairs
- Each square has area 2016/56 = 36, so side 6 (a whole number, good).
- The 56 squares form an m-by-n grid with m*n = 56.
- Non-congruent factor pairs: (1,56),(2,28),(4,14),(7,8), so 4 rectangles.
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