## 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)**

if the question is

If X = P*N^K + P where N and K are positive integers, is X divisible by 2?

then the ans is B.

Could you confirm my answer @egmat help

sorry, but the question given in the PDF document was –>

If X = P*N^K + P where N and K are positive integers, is X divisible by 2?

My thoughts:

the expression reduces to X = P*(N^K+1). For X to be divisible by 2, it’s clear that P needs to be divisible by 2. So, in a way, question is asking us to evaluate “Is P even?”

If P is Even, then P is divisible by 2, hence X is divisible by 2

If P is Odd, then P is not divisible by 2, hence X is not divisible by 2.

Am i misinterpreting the question? OR did the question link from PDF navigate me to incorrect solution webpage?

The option 1 doesn’t even talk about P. it gives values of N + KN = 915. For me, it’s irrelevant info.

The Option 2 says P^35 + 35^P is Even; This is possible only if P is Odd.

so for me, option B is sufficient.

How do we know “P” is an integer ?

Hi Sid,

The questions statement mentions that p, n, and k are positive integers.

Please let us know if you have any further questions.

How do we know P is odd though??

Hi Lynne,

Happy to answer your question.

I presume you are hinting at the following: From Statement (1) we have deduced N to be odd, however, there is no mention of P. In that case, how can we be so certain that X is divisible by 2?

Explanation:

For X to be divisible by 2, we know that X must be even. Consequently, for X to be even, either P must be even or (N+1) must be even.

Note, this is the minimum satisfying condition. If we can establish with certainty that one of the terms (and not necessarily both) is even, it doesn’t matter if the other terms are even or odd.

Let’s take cases to understand this better:

Case 1: P = 2 (E), N = 2 (E); X = 4 (E)

Case 2: P = 2 (E), N = 3 (O); X = 6 (E)

Case 3: P = 3 (O), N = 2 (E); X = 6 (E)

Case 4: P = 3 (O), N = 3 (O); X = 9 (O)

Observe that for X to be even (Cases 1-3), the only necessary condition is for one of the terms to be even.

From Statement (1), since we already know that N is odd, (N+1) will always be even. Hence, irrespective of what P is, X will always be even and we can be certain that X will be divisible by 2.

Hope this clarifies.