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GMAT questions involving fractions: How to solve them quickly

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GMAT fractions

There are some questions involving fractions that certainly seem very difficult at first glance. You need a systematic method to solve such questions.

Consider the following question from The Economist GMAT Tutor's database of practice questions:

If a is an integer and (a2)/(123) is odd, which of the following must be an odd integer?
A) a/4
B) a/12
C) a/27
D) a/36
E) a/72

Tip: try plugging in a value for ‘a’

One approach to this question is to try to work out a possible value for a and use this as a plugin. There are many things to remember as you do this.


  • You are told that (a2)/(123) is odd. Therefore, a2 must be at least as large as 123, or is possibly larger than 123, since the entire expression must be an integer. How large is 123?
  • Break 123 down into its prime factors:


= (3*4)3 
= (3*2*2)3 
= (3*22)3 
= 26 *33


  • a is an integer. Therefore, a2 is the square of an integer, or a perfect square
  • Is it possible that a2 = 26 * 33? For this to be true, the square root of 26 * 33 must be an integer. What is the square root of 26 * 33? This is 23*33/2, which is not an integer. Therefore, a2 cannot be 26 * 33. The problem lies with the fact that 3 is raised to an odd power. If we raise 3 to an even power, we will obtain an integer as the square root.
  • Let us test whether it is possible that a2 = 26 * 34. (We are now multiplying by an additional 3.) The square root of 26 * 34 = 23 * 32 = 8*9 = 72. 72 is an integer. Therefore, it is possible that a = 72.
  • Plug 72 into each of the answer choices:


A) 72/4 = 18 – an even integer
B) 72/12 = 6 – an even integer
C) 72/27 = a non-integer
D) 72/36 = 2 – an even integer
E) 72/72 = 1 – an odd integer

Therefore, E is our answer.

Note the usefulness of prime factorization in solving this question. Although there are many steps in solving such a question, as long as you know the right method, you will be able to get to the correct answer quickly enough.

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