tonebeeze wrote:If a and b are both positive integers, is b^(a+1) - ba^b odd?

(1) a + (a + 4) + (a - 8) + (a + 6) + (a - 10) is odd
(2) b^3 + 3b^2 + 5b + 7 is odd

For even/odd questions, it often helps to draw a table to explore all possibilities.
In the target question, (Is b^(a+1) - ba^b odd?), there are 4 cases to consider:

case a) a=even, b=even
case b) a=odd, b=even
case c) a=even, b=odd
case d) a=odd, b=odd

For each case, let's examine the output of b^(a+1) - ba^b

case a) a=even, b=even --> output is even
case b) a=odd, b=even --> output is even
case c) a=even, b=odd --> output is odd
case d) a=odd, b=odd --> output is even

So, for b^(a+1) - ba^b to be odd, it MUST be the case that a=even and b=odd
So, we can now rewrite our target question as "Is it the case that a is even and b is odd?

Now let's examine the statements.

Statement 1: a + (a + 4) + (a - 8) + (a + 6) + (a - 10) is odd
Simplify left-hand side to get: 5a - 8 is odd
Since 8 is even, we can conclude that 5a must be odd
For 5a to be odd, a MUST be odd
If a must be odd, we can answer the new target question (Is it the case that a is even and b is odd?) with certainty.
As such, statement 1 is SUFFICIENT

Statement 2: b^3 + 3b^2 + 5b + 7 is odd
It's hard to tell if this statement tells us anything about b.
Let's use a table to look at the two possible cases for b
case a) b is odd
case b) b is even

For each case, let's examine the output of b^3 + 3b^2 + 5b + 7
case a) b is odd --> output is even
case b) b is even --> output is odd
Since we are told that b^3 + 3b^2 + 5b + 7 is odd, we know that b MUST be even
If b must be even, we can answer the new target question with certainty.
As such, statement 2 is SUFFICIENT

So, the answer is D