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Strategy for Picking Numbers on "Number property" [#permalink]

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07 Dec 2004, 06:04

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Hi,

I was wondering if anyone has a strategy (step-by-step method) for picking numbers on difficult DS questions involving real numbers/intergers/fractions/inequalities as such. I have seen that once you are at a stage to see if the 2 statements are together sufficient or not (choice C and E) its kind of confusing to pick numbers and you end up losing a lot of time and even in the end you are not so sure if you have covered all possibilities. Given that one is looking to score well on the quantitaive section, such questions are a sure to appear and wanted to know if anyone here as a sure-shot method to cover all possibilities, any views on how to best solve the able type of problems will be much appreciated.

When picking numbers, make sure you read the question carefully. For instance, when question says positive integers, make sure you test only those, not 0, not fractions and negative numbers. This is something that we tend to overlook when solving a question.
As a rule of thumb, there are certain critical values that I tend to plug in:
x<-1
x=-1
-1<x<0
x=0
0<x<1
x=1
x>1
When picking numbers <-1 or >1, don't break your head necessarily with complicated numbers. Stick to say 2 then 10 then 100 and the same with negative numbers. Unless there is some kind of geometric progression in the question, it is not necessary to go through 2,3,4,5 and so on so as to try them all.
Remember that for exponent questions, it is very important to test negative numbers as well as fractions unless the question explicitly states that the range of answers should be positive or should be integers.
This is just a rule of thumb and with practice, you will see what numbers you may need to use in the "plugging in" method. _________________

My approach to plugging in numbers is a little different all together. i think plugging in numbers needs to be part of an overall approach - one of the tools of many to understand a problem.

There are lots of reasons to plug in numbers that should be avoided. If you're trying to avoid learning about positive/negative numbers, or what happens when the numerator is smaller than the denominator, or how evens and odds work, or what to do when 2000 is a factor of 4x, then you're plugging in for the wrong reasons.

You've got to understand those nuances for this test. The more you can think globally about a problem, the less you've got to plug in a number.

Now, in my book, once you do undersatnd those things and their relationships, that's when you can plug in - not to solve the problem, but to make it more tangable in the short run to better understand what's happening.

So if they say x is a multiple of y, don't just plug in two numbers that work. Tell yourself what a multiple is, and figure out what one thing has to do with another. Then choose some numbers that work so you can see what really happens. For example - 24 is a multiple of 6 - tells you that 24/6 is an integer. So x/y must be an integer. Now you've got information and you can take that through the problem. If you're just looking at 24 and 6, that may not be relevant for the problem at hand.

I hope this helps - it's a little disjointed, but it's what I feel is the best approach.

1. Watch for |X|, GMAT likes questions involving modulus.

2. In general, X^^2 is greater than X. However, if X is a fraction between 0 and 1, any higher (n greater than or equal to 2) powers of X ie X^^N < X i.e. X is bounded with in 0 and 1. i.e. 0 < X < 1.

3. Note that 2 is a very special number. It is an even number and also a prime number. When plugging in numbers for a problem involving prime numbers, usually we think of odd numbers but not 2. Note that GMAT is setting up a trap for you.

4. Note that the inequality changes when multiplied by negative numbers.

eg: X < 2 implies -2X > -4

By multiplying by -2, the inequality has changed.
However, multiplying by a positive number, the inequlity does not change.

For example, if X < 2 then 3 X < 6.

5. If the problem involves 3 consecutive integers, you might want to try n-1,n,n+1. Note that the sum of 3 consecutive integers, in this case is 3n. So, you can see right away that irrespetive of what n is, a sum of three consecutive integers is always divisible by 3.

6. Translate information in the question into equations and vice versa.

i.e. As already posted by someone, if y is a factor of x, x/y = integer.
Write this down on the scratch paper so that you know you need to include this information to solve the problem.

Another example is, if the problem says, x and y are positive integers, write it down as x > 0, y > 0.

7. Note the following simple facts:

sum, product or difference of two even numbers yields an even number.

Sum and difference of two odd numbers is Even.
product of two odd numbers is Odd.

sum or difference of an odd and even number is odd.
Product of an odd and even number is Even.

8. Know the difference between a factor and a multiple. Do not get confused.

9. Note that for any real number x not equal to zero, x ^^ 2 is always positive.

10. Even though X^^2 = 16 has two solutions, X^^3 = 8 has only one real solution. i.e. X ^^3 = 8 does not mean x is either 2 or -2. NO. X is 2.

Eg:

1. statement 1 : X^^2 = 4, Statement 2: X^^3 > X Answer C.

11. As already posted earlier, test some critical conditions such as

x=0,1,-1.
x=even and odd
x=fraction between 0 and 1

12. Note that 2 independent equations are needed to solve for x and y. GMAT likes to set up a trap where Statement 1 and Statement 2 appear to provide two equations. So, it is natural to pick answer C thinking that two equations and two unknown. Write down the equations. In some cases, both statements 1 and 2 might give you the same equation. In other words, you will end up with only one equation to solve for x and y. So, in this case the answer should be E.

13. Do not assume any information that is not provided.

For example, GMAT tries to set you up by saying x,y,z are three consecutive integers. This does not mean

In the past, for picking numbers, the following numbers have worked GREAT for me:

-1, -0.5, 0, 0.5, 1

This covers some traps that Srinivas pointed in his post above (square of most numbers are greater than the number itself, unless when the number lies between 0 and 1).

If the question says positive integers, I always try 1, 2, 3. The reason is, again as Srinivas pointed, 2 is a special case (even prime), 3 is an odd prime, and 1 is ALWAYS special!

Today I came across a DS which had to do with inequalities. I'd like to add a tip here: When solving for integer inequalities,e.g. a > b, followed by additions/ subtractions etc, it helps to have four sets of negative and positive numbers to give you all angles of the situation.
e.g. 3 and 5, 3 and -5, -3 and 5 and -3 and -5.

Initially I was making the mistake of plugging only positive intergers. But was not reaching the solution. Now, I make sure that I plug in -ve no's also. Thanks for this post.

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