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How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?

Guest Sep 10, 2017
 #1
avatar+93866 
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How many distinct, non-equilateral triangles with a perimeter of 60 units have integer side lengths $a$, $b$, and $c$ such that $a$, $b$, $c$ is an arithmetic sequence?

 

let the sides by a, a+d, a+2d    where a and d are both positive integers and d>0

 

\(a+a+d+a+2d=60\\ 3a+3d=60\\ a+d=20\\ \)

d a a+d a+2d

Is the sum of the two little numbers

bigger than the 3rd one.

If not then it cannot be a triangle

 

19 1 20 39 no
18 2 20 38 no
17 3      
16 4      
15 5     n
14 6 20 34 no
13 7     n
12 8     n
11 9 20 31 no
10 10 20 30 no
9 11 20   yes  1
8 12 20 28 yes 2
7 13 20 27 y 3
6 14 20 26 4
5 15 20 25 5
4 16 20 24 6
3 17 20 23 7
2 18 20 22 8
1 19 20 21 9

 

9 of the combinations work :)

Melody  Sep 10, 2017

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