each player is given a rectangular piece of graph paper that is 56cm long and 84 cm wide. The horizontal and vertical lines are spaced 1 cm apart. The paper is to be cut along the grid (graph) lines into square pieces that are all the same size without having any paper left over. The winner is the one who cuts the largest square pieces of paper. What would be the length in centimeters of the side of each winning square.
let each side of the largest square be x cm
so along the long side we can fit 84/x squares and along the short side we can fit 56/x squares
56 = 7*4*2 = 28*2
84 = 7*4*3 = 28*3
since 28 is the largest whole number that divides into both 56 and 84, the largest square is
28 x 28 cm.
(there will be 6 of these)
check: 6(28x28) = 4704
56x84 = 4704
To find the length of the side of each winning square, we need to determine the greatest common divisor (GCD) of the length (56cm) and width (84cm) of the rectangular piece of graph paper.
One approach to find the GCD is to list all the factors of both 56 and 84 and identify the largest common factor. However, this can be a time-consuming process.
A more efficient method is to use the Euclidean algorithm to find the GCD. Here's how it works:
Step 1: Divide the larger number (84) by the smaller number (56).
84 ÷ 56 = 1 with a remainder of 28
Step 2: Divide the previous remainder (56) by the current remainder (28).
56 ÷ 28 = 2 with no remainder
Step 3: Since there is no remainder, the current remainder (28) is the GCD.
Therefore, the GCD of 56 and 84 is 28.
Now, we can use the GCD to determine the length of the side of each winning square. Since the GCD represents the largest possible square size that evenly divides both the length and width, we can conclude that each winning square will have a side length of 28 cm.