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?

xXxTenTacion
Jul 13, 2018

#1**+1 **

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

CPhill
Jul 13, 2018