How many squares fit in a rectangle

Find the prime factorization of $2148$ and $1272$ : $\begin{aligned} 2148 &= 2^2 \times 3 \times 179 \\ 1272 &= 2^3 \times 3 \times 53 \end{aligned}$ Thus, the greatest common divisor of the two dimensions is $2^2\times 3 = 12$ . This is the greatest possible side length of a square, assuming a whole number of tiles fit along each side. So there are $\frac{2148}{12}\times\frac{1272}{12}=179 \times 106 = 18974$ total tiles, each with an area of $12\times12=144$ . The answer is $18974+144=\boxed{19118}$ .

To solve the question:

• Find the highest common factor of 2148 and 1272 to find the largest possible side of the square, then find its area.
• Divide the area of the rectangle by the area of the square to find the number of squares that cover the rectangle.
• Add the answers from previous two steps together to answer the question.

Solution: 144 + 18974 = 19118

There are only 6 possible sizes of squares which can cover the whole rectangular area, measuring 2148 cm by 1272 cm, as 2148 and 1272 have only 6 common factors. The possible areas of square are: 1 $*$ 1=1, 2 $*$ 2=4, 3 $*$ 3= 9, 4 $*$ 4=16, 6 $*$ 6=36, 12 $*$ 12=144, in $cm^2$ . They all divide 1272 $*$ 2148.