SOLUTION: Michael is an elective programme student who is working on an assignment. He plans to cover a rectangular sheet of paper of dimensions 126 cm by 108 cm with identical square patter

Algebra ->  Divisibility and Prime Numbers -> SOLUTION: Michael is an elective programme student who is working on an assignment. He plans to cover a rectangular sheet of paper of dimensions 126 cm by 108 cm with identical square patter      Log On


   

Question 1164338: Michael is an elective programme student who is working on an assignment. He plans to cover a rectangular sheet of paper of dimensions 126 cm by 108 cm with identical square patterns.
(i) What is the least number of square patterns that could be formed on the sheet of paper?
(ii) How do you determine what other shapes can the patterns be if they are to fit the sheet of paper perfectly? explain your answer.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(i) least number of squares needed

If the number of squares is to be the least, the size of the squares must be the greatest. The side length of the largest square is the greatest common factor (GCF) of 126 and 108.

My preferred method for finding the GCF of two numbers is this:
(1) make a fraction of the two numbers
(2) simplify the fraction
(3) The GCF is the number you divided by to do the simplification

126%2F108+=+%287%2818%29%29%2F%286%2818%29%29+=+7%2F6

The GCF is 18, so that is the side length of the largest square.

The number of squares required to cover the paper is

%28126%2F18%29%2A%28108%2F18%29+=+7%2A6+=+42

ANSWER: The least number of squares to cover the paper is 42.

(ii) other patterns that can cover the paper....

(iia) other squares

In part (i) we found we could cover the paper with 7*6 = 42 squares with side length 18. We can divide the side length of each of those squares by any factor of 18 to get another way to cover the paper with squares:

18x18 squares: (126/18)(108/18) = 7*6 = 42 squares

Divide the side length by 2:
9x9 squares: (126/9)(108/9) = 14*12 = 168 squares

Divide the side length by 3:
6x6 squares: (126/6)(108/6) = 21*18 = 378 squares

...

Divide the side length by 18:
1x1 squares: (126/1)(108/1) = 13608 squares.

(iib) non-square shapes....

The only other shapes that will exactly cover a rectangular piece of paper are rectangles.

We can obviously cover the paper with 1 rectangle of dimensions 126x108:
126x108 rectangles: # of rectangles = (126/126)(108/108) = 1

Similar to how we found smaller squares that can cover the paper, we can divide the dimensions of the rectangle by any factor of the GCF 18 to get another way to cover the paper.

Divide each side length of the 126x108 rectangle by 2:
(126/2)(108/2) = 63x54 rectangles; # of rectangles 2*2=4

Divide each side length of the 126x108 rectangle by 3:
(126/3)(108/3) = 42x36 rectangles; # of rectangles 3*3=9

...

Divide each side length of the 126x108 rectangle by the GCF 18:
(126/18)(108/18) = 7*6 rectangles; # of rectangles 18*18=324