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(1)+(2) b=3a+3 and b=odd+a --> b=odd+a=3a+3 --> odd=2a+3 --> 2a=even --> a is an integer --> b is an integer. If a is even --> b=3a+3=odd and a-3b=even-odd=odd; If a is odd --> b=3a+3=even and a-3b=odd-even=odd. Sufficient.

An even number is an integer that is "evenly divisible" by 2 i.e., divisible by 2 without remainder, an integer of the form n = 2k, where k is an integer.

An odd number is an integer that is not evenly divisible by 2, an integer of the form n = 2k + 1, where k is an integer.

As for the question: D can not be the answer, see the solution above two different answers possible for (1) and (2).
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