A good question that tests the understanding of Odd-Even properties. The best approach would be to apply the Odd-Even properties rather than plugin values
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?
Let's analyze the data given and list out the possible scenarios.
For
a- b to be an even integer, there are only 2 possible scenarios for a and b.
Case 1: a is even and b is even .
Even - Even = EvenCase 2: a is odd and b is odd.
Odd - Odd = Even It's also given that
a/b is also even. That means a is a multiple of b and the possible cases that satisfy this condition are
Case 1: a is even and b is even
Even/Even can result in an even integer. For eg. 8/2 = 4 (Even)
Case 2: a is even and b is odd.
Even/Odd can also result in an even integer For eg. 6/3 = 2 ( Even)
Combining both conditions, we can conclude that
both a and b has to be even to satisfy both statements.
Do you think this is true in all cases? Let us try it out. 8/2 = 4 (Even) but 6/2 = 3 (Odd)
That means this is not true for all cases. So, we need to be more specific about a.
a/b = Even
a = b* Even = Even * Even as we already found that b is even.
Hence, we can clearly say that
'a' has to be a multiple of 4 as it's a product of two even numbers. For b, we can only say that it's an even integer i.e a multiple of 2.
The question here is
which of the following must be an odd integer?Since it's a
MUST BE type question, we will try to eliminate each answer choice by proving that it could be an even integer.
A. a/2 ==> Since a is a multiple of 4 , a/2 is always even. 8/2 = 4 ,12/2= 6 . Hence, eliminated.
B. b/2 ==> We know that b is an even number, so b/2 can be odd or even depending on the value of b i.e 2/2 = 1 but 4/2 = 2(Even) . Eliminated.
C. (a+b)/2 ==> (a+b ) will be an even integer and Even/2 can be odd or even as explained in option B. So we can eliminate Option C as well.
D. (a + 2)/2 ==> a/2 + 1. Since a is a multiple of 4, a/2 is always even. So, Even + 1 will
always give an odd integer. Therefore, option D is the answer
E. (b+2)/2 ==> b + 2 should be an Even number and Even/2 can be odd or even . Eliminated.
Option D is the correct answer.Thanks,
Clifin J Francis,
GMAT Mentor
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