Problem 19 · 2012 Math Kangaroo
Stretch
Number Theory
divisibilityfactorization
A rectangular piece of paper is 108 mm long and 84 mm wide. After making a straight cut you have a square and a leftover piece. You do the same with the leftover piece and so on until the leftover piece itself is a square. What is the side length of the last square?
Show answer
Answer: E — 12 mm
Show hints
Hint 1 of 2
Each cut peels off the biggest square it can; track the leftover rectangle's sizes.
Still stuck? Show hint 2 →
Hint 2 of 2
This is the subtract-the-smaller-side process, and it ends at the greatest common divisor.
Show solution
Approach: repeated squares (Euclid-style subtraction)
- From 108x84 cut an 84x84 square; leftover is 24x84.
- From 24x84 cut 24x24 squares (three of them); leftover is 24x12.
- From 24x12 cut a 12x12 square, and the leftover is itself a 12x12 square.
- So the last square has side 12 mm.
Mark:
· log in to save