SOLUTION: Suppose you have a rectangle with length 90 units and width 26 units. Each turn, you cut off the greatest possible square from the rectangle. You do so until you have only squares.

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Question 1174876: Suppose you have a rectangle with length 90 units and width 26 units. Each turn, you cut off the greatest possible square from the rectangle. You do so until you have only squares. How many squares will you get?
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

This is one way to draw the diagram

The blue squares are 26 by 26. There are 3 of these blue squares.
The green squares are 12 by 12. There are 2 of these green squares.
The red squares are 2 by 2. There are 6 of these red squares.

Note how 90*26 = 2340
While, 3*26^2+2*12^2+6*2^2 = 2340

This shows that 3*26^2+2*12^2+6*2^2 = 90*26
And it arithmetically backs up what the diagram is showing.

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Here's how I formed that diagram:
First I drew a 90 by 26 rectangle. The rectangle is wider than it is tall, ie it's more flat than tall.

Then I cut three squares that are each 26 by 26 from the right side of the rectangle.
The largest squares we can form at first are this size because any larger than 26 will have the square spill outside the rectangle. We pick on the smaller dimension 26 of the 90 by 26.

After forming those three blue squares, we have a 12 by 26 rectangle to work with.
The 12 is from 90-3*26 = 12
Since 12 < 26, we can't form a fourth blue square.

Focus on the 12 by 26 rectangle. The largest square we can form is 12 by 12. Again we pick on the smaller dimension.
We can form two of these green rectangles because 2*12 = 24 is the largest value smaller than 26. There's no room for a third green square.

Cut away the green squares and you'll be left with a 12 by 2 rectangle. We can perfectly subdivide this into 6 red rectangles that are each 2 by 2.
Note how 6*2 = 12, or you could say that 12/2 = 6


To summarize:
  • we have 3 blue squares that are 26 by 26. So the total blue area is B = 26^2+26^2+26^2 = 2028
  • we have 2 green squares that are 12 by 12. So the total blue area is G = 12^2+12^2 = 288
  • we have 6 red squares that are 2 by 2. So the total red area is R = 6*2^2 = 24
The total area is B+G+R = 2028+288+24 = 2340
This matches with the 90*26 = 2340 to help confirm we have the proper partition

The last thing we need to do is to count how many squares were formed
(3 blue)+(2 green)+(6 red) = 3+2+6 = 11 total squares.
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Answer: 11 squares