Payal Tandon
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## Question P1.2

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

1. N + KN = 915
2. P35 + 35P is Even

A

### Solution

Steps 1 & 2: Understand Question and Draw Inferences

X = P*NK + P

–>  X = P(1 + NK)

X will be odd only if P is odd and (1+NK) is odd

NK 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

P35 + 35P 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.