EXAMPLE: The sum of any two even integers is even. Answer: Proof. (direct) Suppose x and y are even integers. By definition of even, let x = 2k and y = 2j for some integers j and k. Then, x+y=2k+2j = 2(k + j) Note that k+j is an integer because it is the sum of integers. Therefore, by definition of even, x + y is even. (c) For integers x and y, if x - y is odd then r is odd or y is odd.

Q: Let a be an integer such that a = 4 mod 12. Use the definition of mod to prove that a² = 4 mod 12-

A: Solution-:                           a≡4 mod 12a2=(4)2 mod 12     =16 mod  12     =4   mod   12

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