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If $ n $ is odd, then $ 3n $ and $ 5n $ are odd, so $ 3n + 7 $ and $ 5n + 11 $ are even. Hence, $ d = 2 $ is achievable.
If $ n $ is even, both expressions are odd, so $ d = 1 $.
Question: How many lattice points lie on the hyperbola $
Solution: A lattice point is a point with integer coordinates. The hyperbola is given by:
Multiply both sides by $ 400 $ (LCM of 25 and 16) to eliminate denominators:
We now look for integer solutions $ (x, y) $ to this equation. Rearranging:
So $ (5, 0), (-5, 0) $ are solutions.
Try $ d = 25 $: $ x + y = 40 $, many such pairs.
Try $ d = 20 $: $ x + y = 50 $, many such pairs.
Try $ d = 10 $: $ x + y = 100 $, many such pairs.