Question

In question 1, do not use any properties of odd and even numbers outside of
their definitions. However, for questions 2 and 3, you can use the following
parity results without having to prove them each time.
For any integers a and b, the product a.b is odd if and only if a and
b are both odd.
For any integers, a and b, the sum a + b is even if and only if a and b
have the same parity.
For all questions, let x; y 2 Z.
1. Prove that if 11x - y is even, then x and y have the same parity.
2. Prove that 5xy + x + y is odd if and only if x or y is odd.
3. Prove that if 2|3x² - 1, then 4 |x² + 7.

Answer 1

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