Hi all!
Ooops..That "seems" like a GMAT question from the tough segment with an agreed difficulty level of 700 but for someone who has a good grip on the topic of numbers from the Arithmetic this question would be at an intermediate level.
Lets zoom it a little more and we can see that the question tests on your understanding of the properties of evens and odds.
Lets summarize the properties before we deep dive into the solution! 1.
An even number added or subtracted with an even and an odd number added or subtracted with an odd number will ALWAYS generate an even number whereas an even added or subtracted with an odd number will always generate an odd number.
2.
Similarly, an even number multiplied with an odd or even will always generate an even number whereas an odd number multiplied with an odd number will always generate an odd number.
3.
An even number divided by an even number doesnt give us any definite conclusion. For example,6 / 2 = 3 which is an odd number but 4/2 =2 which is an even number. An odd number divided by another odd number will give us an odd value as the result if the result is an integer.
Well hold on now. Knowing this properties is half the story but applying them gives you the GMAT grip on application!
Here is where we would try our hand at the properties on this question!
Since its a DS question, we shall first look at the Q. stem and see if we can break it and simplify further before we proceed to the statements.
We know m and n are positive integers from the Q. stem and so the quick inference is m and n cannot be 0 and belong to the set {1,2,3,4,......}
We are now asked if n is even? So can n be expressed as 2k where k is an integer and/or does n satisfy the properties of the ODD/EVEN nos is the alternate thought process.
Lets now strike at the statements;
St(1) says m(m + 2) + 1 = mnThere are two ways to look at the statement.
First way- Use the properties you just learnt above!
Case 1: If m is odd, then m+2 is also odd(say m=1 then m+2=1+2=3 which is odd). Hence the product m * (m+2) is odd * odd and this will give us
an odd result. m * (m+2) + 1 thus becomes odd + 1 which is even.
Since its an equation, the value mn at the right side of the equation also needs to be EVEN. This gives us an inference that n has to be EVEN for mn to be EVEN with m being odd. The definite answer to the question stem is YES.
Summarizing it ; [O* O ] + 1 = O + 1=mn=E and since m is Odd n has to be even
Case 2:If m is even then m+2 is also even (say m=2 then m+2=4 which is even). Hence the product m * (m+2) is even * even and this will give us an even result (following the properties explained above)
This implies m*(m+2) + 1 = even + 1 =odd
iAs its an equation,t he value mn at the right side of the equation also needs to be ODD.
This is NOT POSSIBLE with m being even and the only possibility is m is ODD as discussed in Case 1 and n is even giving us a definite YES as answer to the question stem.
So we have a definite answer to the Q.stem of a YES from statement 1.
You can right away eliminate options B,C,E as statement 1 now is sufficient.
Lets now attack statement 2.
St(2) says m(m + n) is odd.Lets again take the cases here.
Case 1: m is even and so we have even * (even+ n) is odd which is NOT POSSIBLE as even * any value will always be even.
Case 2 : m is odd and this gives us odd * (odd + n) as odd.
This implies (odd + n ) has to be odd
This again gives us n as even as odd + even is odd.
We again have a definite answer from statement 2. So eliminate answer B and D becomes the appropriate answer choice for this question.
[Refer to attached image]
As a second approach,you can alternately look at statement one and isolate n as n is being asked in the Q.stem.
In that case,
St(1): m(m + 2) + 1 = mn
This gives us n = (m+2) + 1/m
The only possible value for 1/m to be a positive integer is when m =1.
This gives us n = (1+2)+1 = 4 and 4 is even.
So on GMAT we can also isolate variables tested and analyze their properties to save time.
Number system is a lengthy and detailed topic. So practice a lot of official questions from this topic and "think" through as you question yourself "how to i eliminate the answer choices", "which smart and simple value could i use to test" , "how do i break down the stub and make it simpler"!
As you question yourself,you shall gain confidence since you would gain clarity on what not to do!
Hope you have gained some insights here and we could be of help
Devmitra Sen
GMAT Quant SME
Attachments
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