Two rectangles

I cut out two rectangles with 54 cm² and 90 cm². Their sides are expressed in whole centimeters. If I put these rectangles together, I get a rectangle with an area of 144 cm². What dimensions can this large rectangle have? Write all options. Explain your calculation.

Correct answer:

n =  6

Step-by-step explanation:

$$\begin{gathered} 54=ab \\ 90=bc \\ b=90/c \\ 54c=90a \\ 3c=5a \\ a=3,b=18,c=5 \\ a=6,b=9,c=10 \\ a=9,b=6,c=15 \\ a=18,b=3,c=30 \\ a=27,b=2,c=45 \\ a=54,b=1,c=90 \\ S=b\cdot (a+c)\text{cm} \\ {S}_1=18\cdot 8=144{\text{cm}}^2 \\ {S}_2=9\cdot 16=144{\text{cm}}^2 \\ {S}_3=6\cdot 24=144{\text{cm}}^2 \\ {S}_4=3\cdot 48=144{\text{cm}}^2 \\ {S}_5=2\cdot 72=144{\text{cm}}^2 \\ {S}_6=1\cdot 144=144{\text{cm}}^2 \\ n=6 \end{gathered}$$

The equations have the following integer solutions:

54=ab
90=bc

Number of solutions found: 6

a₁=3, b₁=18, c₁=5
a₂=6, b₂=9, c₂=10
a₃=9, b₃=6, c₃=15
a₄=18, b₄=3, c₄=30
a₅=27, b₅=2, c₅=45
a₆=54, b₆=1, c₆=90

Calculated by our Diofant problems and integer equations.

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