Brain Teaser Chapter 4 Group Project

Learning Team Brain Teaser

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Learning Team D Brain Teaser
Nicole Fredrickson, Dede Elliott, Cela White & Tomorrow Nelson

Math 213
July 13, 2015
Zarmina Peracha

Learning Team Brain Teaser

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As individuals, the team explored the question:
For any n x m rectangle such that GCD(n,m) = 1, find a rule for determining the number of unit squares (1 x 1) that a diagonal passes through. The rectangles are a) 2 by 7 and b) 3 by 4.
In order to come up with an appropriate solution, one must consider the greatest common divider and its possibilities. The numbers need to be considered, are they prime or composite numbers? Is the slope a factor in finding the answer?
The first answer provided was to find out the number of squares the diagonal line goes through, first divide the L (length) of the rectangle by the W (width). This will not be a whole number so rounding up necessary, then multiply that number by the W, which provides an answer. For example if the first rectangle W=2 and L=7 when seven is divided by two the answer is 3.5. Round that number up to 4 and then multiply by (2 width) to get 8. If the W=3 and
L=4 then divide four by three and get 1.33 and round up to two. Multiply two by three and get six. This theory could work, however, one must ask how would you write an equation to tell someone to round up. If the answer yielded two even numbers then rounding would not be necessary. On the other hand, rounding may be needed for odd integers.
Another theory to solve this question could be to multiply m by n and divide by two. This doesn’t work because in the 2x7 rectangle the diagonal passes through 8 squares. There are instances when this theory might work, but it is not a valid theory because it is not sure to work every time.
Initially, my first thought on how to solve this problem was to go through with the above theory, (multiply m by n, then divide by two), but this obviously does not work because the diagonal line passes through 8 squares, not 7. By definition, the GCF (greatest common factor), is the greatest factor or number that divides into two numbers. When you begin to look at it as a

Learning Team Brain Teaser

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GCF problem, it works every time because the number of units the line passes through will always have a GCF.
The final theory used to determine the number of unit squares (1 x 1) that a diagonal passes through the rule is m+n­HCF (highest common factor). The number of squares the diagonal passes through when m and n are relatively prime is m+n­1. If a rectangle was 8x4 then the equation would be 8+4­4, so the diagonal would pass through 8 squares because the highest common factor of 8 and 4 is 4. This theory gives an accurate answer on all occasions.