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Updated on: 03 May 2025, 04:35
Originally posted by EgmatQuantExpert on 09 Apr 2015, 06:04.
Last edited by Bunuel on 03 May 2025, 04:35, edited 6 times in total.
Last edited by Bunuel on 03 May 2025, 04:35, edited 6 times in total.
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Is the product of integers M and N even?
(1) N can be expressed as a difference of squares of two consecutive prime numbers at least one of which is odd. M can be expressed as a product of two natural numbers P and Q, where 2P + 1= Q.
(2) N can be expressed as a difference of squares of two consecutive prime numbers which lie at a distance of 2 units. M is the sum of all the numbers from 1 to Z where (Z+1) is a multiple of 4.
This is
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(1) N can be expressed as a difference of squares of two consecutive prime numbers at least one of which is odd. M can be expressed as a product of two natural numbers P and Q, where 2P + 1= Q.
(2) N can be expressed as a difference of squares of two consecutive prime numbers which lie at a distance of 2 units. M is the sum of all the numbers from 1 to Z where (Z+1) is a multiple of 4.
This is
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Updated on: 07 Aug 2018, 05:36
Originally posted by EgmatQuantExpert on 10 Apr 2015, 07:53.
Last edited by EgmatQuantExpert on 07 Aug 2018, 05:36, edited 1 time in total.
Last edited by EgmatQuantExpert on 07 Aug 2018, 05:36, edited 1 time in total.
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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
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
General Discussion
09 Apr 2015, 10:26
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1) For N, we can surely say that it is odd. Since 2 & 3 are the only consecutive prime numbers, N = \(3^2 - 2^2\) = 5 which is odd.
Form M, we know that M = P*Q where Q is odd. But we don't know anything about P. So nature of M can't be found.
So, insufficient.
2) For N, we can surely say that it is even because difference of squares of any two prime numbers which differ by 2 shall always be even.This is because N = \((X+2)^2-X^2 = X^2 + 4 + 2X - X^2\) = 2X + 4 which is even for any number X.
Since N is always even here so product of M & N will be even no matter what M is.
Sufficient.
So B.
Form M, we know that M = P*Q where Q is odd. But we don't know anything about P. So nature of M can't be found.
So, insufficient.
2) For N, we can surely say that it is even because difference of squares of any two prime numbers which differ by 2 shall always be even.This is because N = \((X+2)^2-X^2 = X^2 + 4 + 2X - X^2\) = 2X + 4 which is even for any number X.
Since N is always even here so product of M & N will be even no matter what M is.
Sufficient.
So B.
