https://gmat.economist.com/blog/quantitative/gmat-questions-involving-fractions-how-solve-them-quickly

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:

123

= (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.

The post first appeared on here https://gmat.economist.com/blog/quantitative/gmat-questions-involving-fractions-how-solve-them-quickly

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