## Question P1.2

**If X = P*N ^{K} + P where P, N and K are positive integers, is X odd?**

**N + KN = 915****P**^{35}+ 35^{P}is Even

### Correct Answer

A

### Solution

**Steps 1 & 2: Understand Question and Draw Inferences**

X = P*N^{K} + P

–> X = P(1 + N^{K})

X will be odd only if P is odd and (1+N^{K}) is odd

N^{K }will have the same Even-odd nature as N (Power doesn’t change the even- odd nature of an expression)

–> (1+N) needs to be odd

–> N needs to be even

So, X will be odd only if P is odd and N is even.

So, let’s now look at the given statements to ascertain the Even-Odd nature of P and N.

**Step 3: Analyze Statement 1**

N + KN = 915

–> N(1+K) = 915, which is an odd number

–> The product of 2 numbers is odd only if both the numbers are themselves odd.

–> N is odd

Since we know that N is not even, we can say for sure that X will not be odd.

Therefore, Statement 1 is Sufficient.

**Step 4: Analyze Statement 2**

P^{35} + 35^{P} is even

–> P + 35 is even (Power doesn’t change the even- odd nature of an expression)

–> P is odd

But, we don’t know if N is even or odd. Thus, Statement 2 is not sufficient

**Step 5: Analyze Both Statements Together (if needed)**

We’ve already arrived at a unique answer in Step 3. Therefore, this step is not required.

**Answer: Option (A)**