GRE/GMAT sample questions with solution

Substitution:

1. If  n is an even integer which one of the following is an odd integer?

   a) n²           b)  -2n-4      c) √(n² +2            d)n+1/2    e)2n²-3

       Sol: It is given that n is an even integer. Let n=2

             n² = 2²= 4 is an even integer. So choice a is eliminated.

            -2n-4 = -2*2 -4   = -4-4 = -8 is an even integer. So choice b is eliminated.

          √(n²+2 =√(2²+2 = √6  is not an integer. So choice c is eliminated.

          n+1/2 = 2+1/2 = 3/2 is not an integer. So choice d is eliminated.

          2n²-3 = 2* 2² – 3 = 8 -3 =5 is an odd integer. So choice e is the answer.

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2.  If n is an integer, which of the following CANNOT be an integer?

       a) n-2/2              b)  √n         c) 2/n+1         d) √(n²+3     e) √(1/n²+2)

       Sol : Let n =0 . So n-2/2 = 0-2/2 = -1 which is an integer. So choice a is eliminated.

                √n = √0 =0 which is an integer. So choice b is eliminated.

              2/(n+1) = 2/(0+1) =2 which is an integer. So choice c is eliminated.

            Next, √(n²+3 = √(0²+3)  = √3 which is not an integer . So it may be the answer.

              √(1/n²+2) = √(1/2)  which is not an integer as well .

             So we choose another number say 1. Then √(n²+3) = √4 =2 which is  an integer .

            √(1/n²+2) = √(1/4) . Thus choice e is the answer.

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3. If x, y and z are positive integers such that x< y < z and x + y + z =6, then what is the value of  

     z?

          a) 1       b) 2               c)3             d) 4               e) 5

          Sol :  From the given inequality x<y<z, it is clear that the positive integers x, y and z are

                   different  and are in increasing order of size.

                 Let x=1  y =2 and z= 3.   So 1< 2< 3 and  x+ y+z = 1+2+3 =6

                So z=3. Choice c is the answer.

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4. By how much is the greatest of five consecutive even integers greater than the smallest

      among them?

      a) 1            b) 2               c)4                   d)8                  e) 10

          Sol:    Let the five consecutive even integers be 2, 4, 6, 8, 10

                   Largest no of these =10

                  Smallest no of these = 2

                 Difference =8

                So answer is d.

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5. Which of the following could be an integer?

      a) The  average of two consecutive integers.

      b) The  average of three consecutive integers.

       c) The  average of four consecutive integers.

      d) The  average of six consecutive integers.

      e) The average of 6 and 9

       Sol:   choose any 2 consecutive integers,  say, 1 and 2.

               Average = 1+2/2 =1.5 is not an integer. So choice a is eliminated.

               choose any 3 consecutive integers,  say, 1 ,2 and 3.

               Averge = 1+2+3/3 =6/3 =2 is an integer. So choice b is the answer.

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