Question 1210181: If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will it cross through?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem.
**Understanding the Pattern**
* **3 x 5 Rectangle:**
* The ant travels 3 units horizontally and 5 units vertically.
* The number of squares crossed is 3 + 5 - gcd(3, 5) = 8 - 1 = 7.
* gcd(3,5) is the greatest common divisor of 3 and 5.
* **Generalizing the Pattern:**
* For an m x n rectangle, the number of squares crossed is m + n - gcd(m, n).
**Applying the Pattern to the 15 x 16 Rectangle**
* **15 x 16 Rectangle:**
* The ant travels 15 units horizontally and 16 units vertically.
* gcd(15,16) = 1. since 15 and 16 are relatively prime.
* The number of squares crossed is 15 + 16 - gcd(15, 16) = 15 + 16 - 1 = 30.
**Therefore, the ant will cross through 30 squares in a 15 x 16 rectangle.**
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