Is the product of integers M and N even?

Detailed Solution

Step-I: Given Info

We are given two integers M and N and are asked to find if their product is even.

Step-II: Interpreting the Question Statement

The product of two numbers would be even if at least one of them is even. So, we need to find if either of M and N is even.

Step-III: Statement-I

The statement tells us that M is expressed as a difference of two consecutive prime numbers of which at least one is odd. Two cases are possible:

• We know that there is only one even prime number i.e. 2, so, if one of the prime numbers is 2, the other would be 3, which is odd. Squaring them would not change their even/odd nature. The difference of an even and an odd number would be odd, so N would be odd

• If both the prime numbers are odd, then the difference of their squares would be even (as odd-odd= even). So, N would be even.

From the above two cases, we can’t say with certainty whether N is odd or even.

The statement also tells us that M is a product of P & Q where Q= 2P + 1. We can infer from this that Q is an odd number, but we do not have any information about the even/odd nature of P. So, if P is odd, M would be odd and if P is even, M would be even. Hence, we can’t say with certainty whether M is odd or even.

Since, we don’t know with certainty that either of M, N is even or not, Statement-I is insufficient to answer the question.

Step-IV: Statement-II

Statement-II tells us that N can be expressed as difference of squares of two consecutive prime numbers which lie at a distance of 2 units. We know that all the prime numbers except 2 are odd, since the next prime number after 2 is 3, we can say that 2, 3 are not the consecutive prime numbers (as they lie at a distance of 1 unit). Thus we can conclude that N can be expressed as difference of two odd prime numbers. The difference of two odd numbers will be even.

So, N would be even.

Note here that we don’t need to find the even/odd nature of M because irrespective of the nature of M, the product of M & N would always be even as N is even.

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

Step-V: Combining Statements I & II

Since, we have a unique answer from Statement- I we don’t need to be combine Statements- I & II.
Hence, the correct answer is Option B

Key Takeaways

1. Know the properties of Even-Odd combinations to save the time spent deriving them in the test.
2. There is only 1 even prime number i.e. 2.
3. Odd/Even number raised to any power would not change its even/odd nature


tapas.shyam- when we say that at least one is odd, it means that either both are odd primes or one is odd prime and one is even prime. From the analysis of St-I we can't say with certainty that N is odd/even
Naina1- In statement-II we do not need to calculate the even/odd nature of M once we have established that N is even, as their product would always be even.


Regards
Harsh