If positive integer N = A – B, where A is a positive integer and B is

Detailed Solution

Step-I: Given Info

The statement tells us about a positive integer \(N = A – B\) and we are told that \(A\) is a positive integer and \(B\) is a prime number. We are asked to find if \(N\) is odd.

Step-II: Interpreting the Question Statement

The expression \(N\) is expressed as a difference of two numbers, for \(N\) to be odd the nature of these two numbers would have to be opposite i.e. one has to be odd and other has to be even. We are given that \(B\) is prime number, we know that except for 2, all the prime numbers are odd. So, if we can establish if \(B\) is greater than 2, we would be able to say with certainty that \(B\) is odd.

Step-III: Statement-I

Statement-I tells us that \(B\) and \(X\) are prime numbers such that \(B-X = 4\), the difference of two prime numbers is even, which would mean that either both are odd or both are even. Since there is only even prime number possible (i.e. 2), we can say with certainty that both \(B\) and \(X\) are odd.

Since \(A\) has only \(B\) and \(X\) as its prime factors, this would imply that \(X\) is a product of two odd numbers. Thus A would also be odd.
We now know the even/odd nature of both \(A\) and \(B\), thus we can determine with certainty the even/odd nature of \(N\).

Hence, Statement-I is sufficient to answer the question.

Step-IV: Statement-II

Statement-II tells us that \(A\) has 9 factors. Since 9 can factorized only as (9*1) or (3*3) this would imply that \(A\) can be written as:

\(A = P_{1}^8 or P_{1}^2*P_{2}^2\), where \(P_{1}*P_{2}\) are the prime factors of \(A\).

Since we are told that \(A\) is divisible by \(B^2\), this would mean that \(P_{1} = B\). Please note here that \(B^2\) is a divisor of \(A\) in both the cases where \(A = P_{1}^8 or P_{1}^2 * P_{2}^2\). So, we can’t say with certainty that \(A\) can be expressed in one of the ways. Let’s evaluate both the cases now:

• Case-I: \(A= B^8\)
In this case \(A\) will have the same even/odd nature as that of \(B\). We know that the difference of two even numbers or two odd numbers would always be even. Hence, we can say that \(N\) will be even.

• Case-II: \(A= B^2*P_{2}^2\)
In this case if \(B\) is an even prime number (i.e. 2) then \(A\) would also be even and subsequently \(N\) would also be even. However if \(B\) is odd, the nature of \(A\) would depend on the nature of \(P_{2}\).

 If \(P_{2}\) is even, then \(A\) would be even and \(N\) would be odd ( since \(B\) is odd)
 If \(P_{2}\) is odd, then \(A\) would be odd and \(N\) would be even ( since \(B\) is odd)

So, we see here that we can’t predict with certainty the exact even/odd nature of \(N\).

Hence, Statement-II is insufficient to answer the question.

Step-V: Combining Statements I & II

Since, we have received our unique answer from Statement-I, we don’t need to combine the inferences from Statement-I & II.
Hence, the correct Answer is Option A.

Key Takeaways

1. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations.
2. Know the properties of Even-Odd combinations to save the time spent deriving them in the test
3. Remember that there is only 1 even prime number i.e. 2.
4. If 2 is a prime factor of the number, the number would be even else it would be odd.
5. If the total number of factors of a number is given to be T, evaluate the possible ways in which T can be expressed as a product of 2 or more numbers.

Regards
Harsh