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(a) Prove that if y = 4k for k ≥ 1, then there exists a primitive...

(a) Prove that if y = 4k for k ≥ 1, then there exists a primitive Pythagorean triple (x, y, z) containing y.

(b) Prove that if x = 2k+1 is any odd positive integer greater than 1, then there exists a primitive Pythagorean triple (x, y, z) containing x.

(c) Find primitive Pythagorean triples (x, y, z) for each of z = 25, 65, 85. Then show that there is no primitive Pythagorean triple (x, y, z) with z = 45 or z = 105. (Show all necessary calculations.)

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