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Updated on: 13 Aug 2018, 03:10
Originally posted by EgmatQuantExpert on 08 Apr 2015, 03:50.
Last edited by EgmatQuantExpert on 13 Aug 2018, 03:10, edited 8 times in total.
Last edited by EgmatQuantExpert on 13 Aug 2018, 03:10, edited 8 times in total.
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Is the product of two integers \(A\) and \(B\) odd?
(1) \(A\) is the number of factors of \(N\), where \(N\) is a perfect square and \(B = A^3 - 1\)
(2) \(A\) is a product of two consecutive prime numbers and when \(B + 3^{11}\) is added to \(A\), the sum is an odd number.
Source: e-GMAT
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(1) \(A\) is the number of factors of \(N\), where \(N\) is a perfect square and \(B = A^3 - 1\)
(2) \(A\) is a product of two consecutive prime numbers and when \(B + 3^{11}\) is added to \(A\), the sum is an odd number.
Source: e-GMAT
This is
Ques 1 of The E-GMAT Number Properties Knockout
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Updated on: 07 Aug 2018, 01:01
Originally posted by EgmatQuantExpert on 10 Apr 2015, 05:40.
Last edited by EgmatQuantExpert on 07 Aug 2018, 01:01, edited 3 times in total.
Last edited by EgmatQuantExpert on 07 Aug 2018, 01:01, edited 3 times in total.
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Detailed Solution
Step-I: Given Info
The question tells us about two integers \(A\) & \(B\) and asks us if their product is odd.
Step-II: Interpreting the Question Statement
To find if the product of two numbers is odd/even, we need to establish if either of the number is even or not. In case either of the number is even, the product would be even else the product would be odd.
Step-III: Statement-I
Statement- I tells us that \(A\) is the number of factors of a perfect square, since the no. of factors of a perfect square is odd, we can deduce that \(A\) is odd. Since \(A\) is not even, to find the nature of product of \(A\) & \(B\), we need to find if \(B\) is odd/even.
It’s given that \(B= A^3 -1\), since we have established that \(A\) is odd, \(A^ 3\) will also be odd. Subtracting 1 from an odd number will give us an even number, hence we can deduce that \(B\) is even.
Since, we know now that \(B\) is even it is sufficient for us to deduce that the product of \(A\) & \(B\) would be even.
Thus Statement-I is sufficient to get the answer.
Step-IV: Statement-II
Statement- II tells us that \(A\) is a product of two consecutive prime numbers, so \(A\) may be even if one of the prime number is 2 and may be odd if none of the prime number is 2. So, we can’t establish with certainty the even/odd nature of \(A\).
The statement also tells us that \(A+ B + 3^{11} =\) odd, since 3 is an odd number, \(3^ {11}\) would also be odd and subtracting an odd from an odd number would give us an even number. So, we can rewrite
\(A+ B =\) even which would imply that either both \(A\), \(B\) are even or both are odd. In both the cases the nature of product of \(A\) & \(B\) can’t be established with certainty, it will be even if both \(A\) & \(B\) are even and will be odd if both \(A\) & \(B\) are odd.
So, Statement- II is not 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 A
Key Takeaways
1. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations.
2.The number of factors of a perfect square would always be odd.
3.Know the properties of Even-Odd combinations to save the time spent deriving them in the test
Regards
Harsh
Step-I: Given Info
The question tells us about two integers \(A\) & \(B\) and asks us if their product is odd.
Step-II: Interpreting the Question Statement
To find if the product of two numbers is odd/even, we need to establish if either of the number is even or not. In case either of the number is even, the product would be even else the product would be odd.
Step-III: Statement-I
Statement- I tells us that \(A\) is the number of factors of a perfect square, since the no. of factors of a perfect square is odd, we can deduce that \(A\) is odd. Since \(A\) is not even, to find the nature of product of \(A\) & \(B\), we need to find if \(B\) is odd/even.
It’s given that \(B= A^3 -1\), since we have established that \(A\) is odd, \(A^ 3\) will also be odd. Subtracting 1 from an odd number will give us an even number, hence we can deduce that \(B\) is even.
Since, we know now that \(B\) is even it is sufficient for us to deduce that the product of \(A\) & \(B\) would be even.
Thus Statement-I is sufficient to get the answer.
Step-IV: Statement-II
Statement- II tells us that \(A\) is a product of two consecutive prime numbers, so \(A\) may be even if one of the prime number is 2 and may be odd if none of the prime number is 2. So, we can’t establish with certainty the even/odd nature of \(A\).
The statement also tells us that \(A+ B + 3^{11} =\) odd, since 3 is an odd number, \(3^ {11}\) would also be odd and subtracting an odd from an odd number would give us an even number. So, we can rewrite
\(A+ B =\) even which would imply that either both \(A\), \(B\) are even or both are odd. In both the cases the nature of product of \(A\) & \(B\) can’t be established with certainty, it will be even if both \(A\) & \(B\) are even and will be odd if both \(A\) & \(B\) are odd.
So, Statement- II is not 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 A
Key Takeaways
1. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations.
2.The number of factors of a perfect square would always be odd.
3.Know the properties of Even-Odd combinations to save the time spent deriving them in the test
Regards
Harsh
Updated on: 07 Aug 2018, 01:03
Originally posted by EgmatQuantExpert on 19 Jun 2015, 21:52.
Last edited by EgmatQuantExpert on 07 Aug 2018, 01:03, edited 2 times in total.
Last edited by EgmatQuantExpert on 07 Aug 2018, 01:03, edited 2 times in total.
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riyazgilani
Dear riyazgilani
Please note that there is a difference between 'consecutive numbers that are prime' and 'consecutive prime numbers.'
Our concept file mentions that 2 and 3 are the only 'consecutive numbers that are prime'
On the other hand, examples of 'consecutive prime numbers' are many: 2, 3, 5, 7, 11, 13, 17, 19 . . . all these prime numbers are consecutive prime numbers, because they come one after the other in the list of prime numbers.
So, when this question says that two numbers are consecutive prime numbers, they can be (2,3) or (5, 7) or (7,11) etc.
I hope this clarification helped.
Best Regards
Japinder
