Is a - 3b an even number?

Is a-3b an even number?


First note that we are not told that a and b are integers.


(1) \(b = 3a + 3\) --> \(a - 3b = -8a - 9\).

\(a = 1\) --> \(-8a - 9 = -17 = odd\).

\(a = \frac{3}{8}\) --> \(-8a - 9 = -12 = even\).

Not sufficient.


(2) \(b - a = odd\) --> \(a = b - odd\) --> \(a - 3b = b - odd - 3b = -2b - odd\)

\(b = odd\) --> \(a - 3b = -2b - odd = odd\)

\(b = \frac{3}{2}\) --> \(a - 3b = -2b - odd = even\)

Not sufficient.


(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.


Answer: C.