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# hw8 - ematical induction 4 Show that if x y z is a...

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Math 465 Problem Set 8 Due Friday, April 18, 2008 1. Find all integer solutions of the Diophantine equation 40 x + 25 y = 600 . 2. Find all integer solutions of the Diophantine equation 8 x 1 - 4 x 2 + 6 x 3 = 6 3. Let a 1 x 1 + a 2 x 2 + ··· + a n x n = b be a linear Diophantine equation in the n ( 2) variables x 1 , x 2 , . . . , x n , and let d be the gcd of a 1 , a 2 , . . . , a n . (a) If d - b , prove that the equation have no solutions. (b) If d | b , prove that the equation have inﬁnitely many solutions. (Hint: Use math-
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Unformatted text preview: ematical induction.) 4. Show that if { x, y, z } is a primitive Pythagorean triple, then either x or y is divisible by 3. (Hint: Use the fact that a 2 â‰¡ 1 (mod 3) if a 6â‰¡ 0 (mod 3).) 5. Find all primitive Pythagorean triples { x, y, z } with z â‰¤ 20. 1...
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