Generate six pythagorean triples using any method discussed during class. explain each method you use.

1) (3,4,5)

2) (6,8,10)

3) (8,15,17)

4) (10,24,26)

5) (5,12,13)

6) (33,56,65)

Step-by-step explanation:

There are many ways of generating Pythagorean triples (a,b,c), one of the easiest ones is to take two natural numbers n and m, and we will have:

a = n² - m²,

b = 2nm,

c = n² + m²

Then (a, b, c) is a Pythagorean triple.

So, we're going to generate 6 Pythagorean triples using this method.

1. m= 1 and n = 2

⇒a = 2² - 1² = 3

b =2(1)(2) = 4

c= 1² + 2² = 5

and the triple is (3,4,5)

2. Now for m = 1 and n = 3

⇒a = 3²-1² = 8

b= 2(3)(1) =6

c = 1²+3² = 10

and the triple is (6,8,10)

3. Now for m = 1 and n = 4

⇒a = 4²-1²=15

b=2(1)(4)=8

c= 4²+1²=17

And the triple is (8,15,17)

4. Now for m = 1 and n =5

⇒a = 5²-1²=24

b=2(1)(5)=10

c= 5²+1²=26

and the triple is (10,24,26)

5. m = 2 n = 3

⇒a = 3²-2²= 5

b= 2(2)(3)=12

c= 3²+2²= 13

And the triple is (5,12,13)

6. m= 4 n = 7

⇒a= 7²-4²=33

b=2(7)(4)=56

c= 4²+7²= 65

And the triple is (33,56,65)

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