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How many non-congruent right triangles are there, all of whose sides have positive integer lengths, and one of whose legs (i.e. not the hypotenuse) has length 162?

 Jul 13, 2018
 #1
avatar+107493 
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One place to start is to  look at the divisors of 162  and see which ones are possible legs for  "primitive" Pythagorean Triple  right triangles

 

The  divisors  of  162  are   1 | 2 | 3 | 6 | 9 | 18 | 27 | 54 | 81 | 162

 

  3   will  work  because   3-4-5  is a primitive  triple

We need to divide  162  by each factor....and then  multiply  both 4 and 5 by this factor to find one triple

 

So

162/3  =   54

So  54 * 4  = 216

And 54 * 5 = 270

So    162 -  216 - 270  is a   "triple"

 

And

9  will work  because

9 - 40 - 41 is a primitive triple

So 162/9  = 18

And  18 * 40  = 720

And 18 * 41  =  738

So  162 - 720 - 738  is a  "triple'

 

There are two more which are a lttle hard to find...the next is

27 - 364 - 365

So 162/27  = 6

And 364 * 6  = 2184

And 365 * 6  = 2190

So 162 - 2184  -2190   is  a "triple'

 

The last  is 162 - 6560- 6562

 

So....there are 4    which have one leg  = 162

 

 

cool cool cool

 Jul 13, 2018

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