If a b, is (a + b) an integer?

Question

GMATbuster's Weekly Quant Quiz#12 Ques #4 
 For Questions from earlier quizzes:  Click Here

If a > b, is (a + b) an integer?
Statement 1: (a - b) is an integer
Statement 2: a + 2b is an integer

ooter · Answer

1) the operation of addition and subtraction are similar and preserve integer property.
2) operation of multiplication may change integer property

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • a > b 
We need to find whether a + b is an integer or not.
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  (a - b) is an integer.
 • If a and b are integers, then a – b will be integer
 o a + b will also be integer. 
• If a = 1.4 and b = 0.4, then a – b = 1, which is an integer.
 o a + b = 1.4 + 0.4 = 1.8 is not an integer.  
We are not getting unique results.
Hence, statement 1 is not sufficient, we can eliminate answer options A and D
 Statement 2:  a + 2b is an integer.
 • If a and b are integers, then a + 2b is an integer
 o a + b is also an integer. 
• If a = 0.8 and b = 0.1, then a + 2b = 1.0 is an integer
 o a + b = 0.9, is not an integer.  
We are not getting unique answer.
Hence, statement 2 is also not sufficient, we can eliminate answer options B.
 Step 3: Analyse Statements by combining. 
From statement 1: a – b is an integer
From statement 2: a + 2b is an integer
By combining the two, 
 • If a and b are integers, then a + b is an integer.
• If a and b are non-integers, for example:  a =\frac{5}{3}  and  b = \frac{-1}{3} , 
 o In this case also,  a - b = \frac{5}{3} - (-\frac{1}{3}) = 2 (Integer) and a + 2b =  \frac{5}{3}  +  2*(\frac{-1}{3})= 1 (Integer) 
 o However,  a + b = \frac{5}{3} + \frac{-1}{3} =\frac{4}{3} , is a fraction  
So, a + b may or may not be an integer. Since, we are not getting a unique answer, the correct answer is  Option E.

mehro023 · Answer

GMATWhizard

it says a>b, how are we taking a and b to have the same value then ?

GMATWhizTeam · Answer

Hey,

Thanks for pointing this out. If a > b, then we cannot consider the same value for a and b. I have edited the solution.

There is an alternate method that doesn't require us to take values and may help in avoiding such mistakes. (I am just showing the third step)

Step 3: Analyse Statements by combining. 
From statement 1: a – b = n, where n is an integer .....(i)
From statement 2: a + 2b =m, where m is an integer.......(ii)

• We can multiply the second equation by 2 and add equation (i) and (ii)
 o On multiplying equation (ii) by 2, we get 2a + 4b = 2m
o On adding it with equation (i) which is a – b = n, we get:
   3a + 3b = (2m + n) 
  a + b = \frac{(2m + n)}{3}   
• Now, we know don’t know if 2m + n is divisible by 3 or not. Hence, we cannot infer if a + b is an integer. 
Hence, the correct answer is  Option E .

GMATinsight · Answer

GMATinsight's Solution

Question : Is (a + b) an integer?

Hint:    always question the information given in question e.g. in the given question the information about a and b to be integer is absent

Statement 1 : (a - b) is an integer

case 1: a = 5 and b = 2, a+b = Integer
case 2: a = 5.2 and b = 2.2, a+b ≠ Integer

NOT SUFFICIENT

Statement 2 : a + 2b is an integer

case 1: a = 5 and b = 2, a+b = Integer
case 2: a = 5.8 and b = 1.1, a+b ≠ Integer

NOT SUFFICIENT

Combining the two statements

a-b = Integer -------(1)and 
a+2b = Integer-------(2)

Taking difference of the two equations i.e. (2) - (1)
3b = Integer

Taking Sum of the two equations i.e. (2) + 2*(1)
3a = Integer

i.e. 3a+3b = Integer
But a+b = Integer /3
Which may or may NOT be an integer hence

e.g.
case 1: a = 6 and b = 3, a-b and a+2b are both integer and a+b = Integer
case 2: a = 5/3 and b = 2/3, a-b and a+2b are both integer BUT a+b ≠ Integer

Answer: Option E

If a > b, is (a + b) an integer?

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