EXERCISE 2.7.2: Proofs by cases - even/odd integers and divisibility.

Prove each statement.

(a) Ifa is an integer, then x2 + 5x - 1 is odd.

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(b) If integers x and y have the same parity, then x +y is even.

The parity of a number tells whether the number is odd or even. If x and y have the same parity, they are either both

even or both odd.

(c)

If integers x and y where x < y are consecutive, then they have opposite parity.

(d) For integers x and y, if xy is odd, then x is odd and y is odd.

(e) If x and y are integers such that a³ (y + 5) is odd, then x is odd and y is even.

(f) Let x and y be two integers. If xy is not an integer multiple of 5, then neither a nor y is an integer multiple of 5.

(g) If x and y are two numbers such that xy and x+y are both even, then x and y are both even.

(h) If n is an odd integer then 81(n²-1).