A rectangle game board measures 126 cm by 90 cm. The game board is divided into small squares of equal size.

Question

Find the longest possible length of the side of one of the small squares.  And find the least number of squares the game board can be equally divided into.

Answer 1

The "least number of squares" means the "largest possible square size."  This is the Greatest Common Factor (GCF) of 126 and 90.

 

And, since you mentioned "Prime Factorization," let's do that.  The prime factors are:

   126:      2*3*3*7

    90:       2*3*3*5

 

The number 2*3*3 =18 contains the most common factors of 126 and 90.  That means that 7 (that is, 126/18) squares by 5 squares (that is, 90/18), total of 35 squares, is the least number of squared that will fit.

Answer 2

The question asks about squares of equal size.  This is equivalent to asking about the gcf of 126 and 90, which is 18.

(You may have learned the Euclidean algorithm to find the gcf; 126 / 90 = 1 r 36; 90/36 = 2 r 18; 36/18 = 2 r 0. You stop when you get a remainder of 0, and 18 is thus the gcf; you can also get the gcf by looking at the prime factorization of the 2 numbers and taking the minimum power of each prime and multiplying - 126 = 2 * 3^2*7; 90 = 2*3^2*5; thus, the gcf is 2*3^2 = 18).  If the squares are 18 x 18, we divide the rectangle into 126/18 * 90/18 = 7*5 = 35 squares. 

Answer 3

We find the greatest common factor of 90 and 126, since the square's side measures add up to 90cm in one dimension and 126 in the other dimension, and that a square has sides of equal measure.

 

List all the common factors

 

90: 1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90

 

126: 1 , 2 , 3 , 6 , 7 , 9 , 18

 

 

At this point, we can stop listing all the factors of 126, since there no more factors between 18 and 90.

 

The greatest common factor is 18.  Therefore, 18 is the longest side the square can have.

Answer 4

Since the board is formed of equal squares the difference between the height and width must represent a whole number of squares  126-90 = 36.       36 cannot be the height of one square because 90 is not divisible by 36.   36 could be the height of two squares since 90/18 =5 and 126/18 = 7.   A square of size 18 is the largest that fits evenly into both the width and height of the rectangle and will produce the smallest number of squares on the game board.

 

So the smallest possible number of squares is 5 times 7 or 35.    We can of course use smaller squares such as 6 or 9 units or even 1 unit each, but those choices would produce a larger number of squares and the question asks for the smallest.