Question

Suppose a teacher has a rectangular sheet of cardboard 420 centimeters long and 378 centimeters wide and that he wants to cut that sheet into many squares, all of the same size. a. what are the dimensions of the largest possible square (whose length is a whole number) that will create no waste?

Answer 1

The dimensions of the largest possible square that a teacher can cut from a 420 cm by 378 cm rectangular sheet of cardboard with no waste are 42 cm by 42 cm, as 42 is the greatest common divisor of the two dimensions.

Step-by-step explanation:

Finding the Largest Possible Square Without Waste

To solve the student's question about cutting the cardboard into squares with no waste, we must find the greatest common divisor (GCD) of the cardboard's length and width. The cardboard measures 420 centimeters long and 378 centimeters wide. The largest square that could be made from the cardboard without any waste would be equal to the GCD of these two dimensions.

To find the GCD of 420 and 378, we can use the Euclidean algorithm:

1. 
   
3. Divide 420 by 378, which gives a quotient of 1 and a remainder of 42.
4. 
   
6. Then divide 378 by the remainder 42, which gives a quotient of 9 and a remainder of 0, indicating that 42 is the GCD.

Therefore, the dimensions of the largest possible square would be 42 centimeters by 42 centimeters.